How many questions from Vector Algebra appear in JEE?

Vector Algebra has appeared in 173 of the last 186 JEE Main and JEE Advanced papers — about 93% of them — contributing 245 questions in total across those papers.

Is Vector Algebra an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 93% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Vector Algebra questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Vector Algebra JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

For real numbers

\alpha, \beta, \gamma, \delta

and

\mu

, consider the matrix

M = \begin{bmatrix} \alpha & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \beta & \frac{1}{\sqrt{3}} \\ \gamma & \delta & \mu \end{bmatrix}

. Suppose that

MM^T = I

, where

M^T

is the transpose of the matrix

M

, and

I

is the

3 \times 3

identity matrix. Let

\vec{u} = \alpha\,\hat{i} + \frac{1}{\sqrt{3}}\,\hat{j} + \gamma\,\hat{k}, \vec{v} = \frac{1}{\sqrt{2}}\,\hat{i} + \beta\,\hat{j} + \delta\,\hat{k}

and

\vec{w} = -\frac{1}{\sqrt{2}}\,\hat{i} + \frac{1}{\sqrt{3}}\,\hat{j} + \mu\,\hat{k}

. Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
P.The value of $\gamma^2 + \delta^2$ is	1.0
Q.If $x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$ for some real numbers $x, y$ and $z$ , then the value of $x$ is	2.1
R.The value of $\|\vec{u} \cdot (\vec{v} \times \vec{w})\|$ is	3. $\frac{1}{\sqrt{2}}$
S.The value of $\|\vec{u} \times (\vec{v} \times \vec{w})\|$ is	4. $\frac{1}{\sqrt{3}}$
	5. $\frac{5}{6}$

(A)(P) → (5), (Q) → (4), (R) → (2), (S) → (1)
(B)(P) → (4), (Q) → (5), (R) → (1), (S) → (2)
(C)(P) → (5), (Q) → (3), (R) → (2), (S) → (1)
(D)(P) → (5), (Q) → (4), (R) → (1), (S) → (2)

Q2·MathematicsSingle correctJEE Advanced 2026

Let

\vec{a}, \vec{b}

be two vectors, and let

P, Q

and

R

be the points with position vectors

\vec{a}

,

\vec{b}

and

\vec{a} + \vec{b}

, respectively, with respect to the origin

O

. If

|\vec{a} + \vec{b}| = \sqrt{21}

,

|\vec{a} - \vec{b}| = 3

, and

\vec{a}

and

(\vec{a} - \vec{b})

are perpendicular to each other, then the area of the triangle

OPR

is

(A) $\sqrt{3}$
(B) $\dfrac{\sqrt{3}}{2}$
(C) $\dfrac{3\sqrt{3}}{2}$
(D) $\dfrac{3}{2}$

Q3·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}

,

\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}

and a vector

\vec{c}

be such that

2(\vec{a} \times \vec{b}) + 3(\vec{b} \times \vec{c}) = \vec{0}

. If

\vec{a} \cdot \vec{c} = 15

, then

\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})

is equal to:

(A) $-6$
(B) $-5$
(C) $-4$
(D) $-3$

Q4·MathematicsNumericalJEE Main 2026

If

\vec{a} = \hat{i} + \hat{j} + \hat{k}

,

\vec{b} = \hat{j} - \hat{k}

and

\vec{c}

be three vectors such that

\vec{a} \times \vec{c} = \vec{b}

and

\vec{a} \cdot \vec{c} = 3

, then

\vec{c} \cdot (\vec{a} - 2\vec{b})

is equal to ______.

Q5·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = 2\hat{i} + 3\hat{j} + 3\hat{k}

and

\vec{b} = 6\hat{i} + 3\hat{j} + 3\hat{k}

. Then the square of the area of the triangle with adjacent sides determined by the vectors

(2\vec{a} + 3\vec{b})

and

(\vec{a} - \vec{b})

is :

(A)450
(B)900
(C)1800
(D)2400

Q6·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = \sqrt{7}\hat{i} + \hat{j} - \hat{k}

and

\vec{b} = \hat{j} + 2\hat{k}

. If

\vec{r}

is a vector such that

\vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0}

and

\vec{r} \cdot \vec{a} = 0

, then

|3\vec{r}|^2

is equal to:

(A)44
(B)54
(C)86
(D)132

Q7·MathematicsSingle correctJEE Main 2026

Let O be the origin,

\vec{OP} = \vec{a}

and

\vec{OQ} = \vec{b}

. If R is the point on

\vec{OP}

such that

\vec{OP} = 5\vec{OR}

, and M is the point such that

\vec{OQ} = 5\vec{RM}

, then

\vec{PM}

is equal to :

(A) $\frac{1}{5}(\vec{a} - 4\vec{b})$
(B) $\frac{1}{5}(\vec{b} - 4\vec{a})$
(C) $\frac{1}{5}(-\vec{a} + 4\vec{b})$
(D) $\frac{1}{5}(-\vec{b} + 4\vec{a})$

Q8·MathematicsSingle correctJEE Main 2026

Let

\hat{u}

and

\hat{v}

be unit vectors inclined at an acute angle such that

|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}

. If

\vec{A} = \lambda\hat{u} + \hat{v} + (\hat{u} \times \hat{v})

, then λ is equal to:

(A) $\frac{4}{3}(\vec{A} \cdot \hat{u}) - \frac{2}{3}(\vec{A} \cdot \hat{v})$
(B) $\frac{2}{3}(\vec{A} \cdot \hat{u}) - \frac{1}{3}(\vec{A} \cdot \hat{v})$
(C) $\frac{4}{3}(\vec{A} \cdot \hat{u}) + \frac{2}{3}(\vec{A} \cdot \hat{v})$
(D) $(\vec{A} \cdot \hat{u}) - \frac{1}{2}(\vec{A} \cdot \hat{v})$

Q9·MathematicsSingle correctJEE Main 2026

Two adjacent sides of a parallelogram PQRS are given by

\vec{PQ} = \hat{j} + \hat{k}

and

\vec{PS} = \hat{i} - \hat{j}

. If the side PS is rotated about the point P by an acute angle

\alpha

in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then

\sin^2\left(\frac{5\alpha}{2}\right) - \sin^2\left(\frac{\alpha}{2}\right)

is equal to:

(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\frac{\sqrt{3}}{4}$
(D) $\frac{2\sqrt{3}}{5}$

Q10·MathematicsSingle correctJEE Main 2026

Let the vectors

\vec{a} = -\hat{i} + \hat{j} + 3\hat{k}

and

\vec{b} = \hat{i} + 3\hat{j} + \hat{k}

. For some

\lambda, \mu \in \mathbb{R}

, let

\vec{c} = \lambda\vec{a} + \mu\vec{b}

. If

\vec{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10

and

\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2

, then

|\vec{c}|^2

is equal to:

(A) $8$
(B) $12$
(C) $14$
(D) $15$

Q11·MathematicsSingle correctJEE Main 2026

If

\vec{a}

and

\vec{b}

are two vectors such that

|\vec{a}| = 2

and

|\vec{b}| = 3

, then the maximum value of

3\left|\left(3\vec{a} + 2\vec{b}\right)\right| + 4\left|\left(3\vec{a} - 2\vec{b}\right)\right|

is :

(A)30
(B)36
(C)60
(D)72

Q12·MathematicsSingle correctJEE Main 2026

For three unit vectors

\vec{a}, \vec{b}, \vec{c}

satisfying

\left|\vec{a}-\vec{b}\right|^{2} + \left|\vec{b}-\vec{c}\right|^{2} + \left|\vec{c}-\vec{a}\right|^{2} = 9

and

\left|2\vec{a} + k\vec{b} + k\vec{c}\right| = 3

, the positive value of k is :

(A)3
(B)6
(C)4
(D)5

Q13·MathematicsSingle correctJEE Main 2026

Let P be a point in the plane of the vector

\overrightarrow{AB} = 3\hat{i} + \hat{j} - \hat{k}

and

\overrightarrow{AC} = \hat{i} - \hat{j} + 3\hat{k}

such that P is equidistant from the lines AB and AC. If

\left|\overrightarrow{AP}\right| = \frac{\sqrt{5}}{2}

, then the area of the triangle ABP is :

(A)2
(B) $\frac{3}{2}$
(C) $\frac{\sqrt{30}}{4}$
(D) $\frac{\sqrt{26}}{4}$

Q14·MathematicsSingle correctJEE Main 2026

Let

\vec{a}=2\hat{i}+\hat{j}-2\hat{k}

,

\vec{b}=\hat{i}+\hat{j}

and

\vec{c}=\vec{a}\times\vec{b}

. Let

\vec{d}

be a vector such that

\left|\vec{d}-\vec{a}\right|=\sqrt{11},\left|\vec{c}\times\vec{d}\right|=3

and the angle between

\vec{c}

and

\vec{d}

is

\frac{\pi}{4}

. Then

\vec{a}\cdot\vec{d}

is equal to

(A) $11$
(B) $3$
(C) $0$
(D) $1$

Q15·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = 2\hat{i} - 5\hat{j} + 5k

and

\vec{b} = \hat{i} - \hat{j} + 3k

. If

\vec{c}

is a vector such that

2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0}

and

(\vec{a} - \vec{b}) \cdot \vec{c} = -97

, then

\left| \vec{c} \times k \right|^{2}

is equal to

(A)193
(B)233
(C)218
(D)205

Q16·MathematicsSingle correctJEE Main 2026

Let

\vec{a}, \vec{b}, \vec{c}

be three vectors such that

\vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c})

. If

|\vec{a}| = 1, |\vec{b}| = 4

,

|\vec{c}| = 2

, and the angle between

\vec{b}

and

\vec{c}

is 60°, then

|\vec{a}.\vec{c}|

is :

(A)2
(B)4
(C)0
(D)1

Q17·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = -\hat{i} + \hat{j} + 2\hat{k}

,

\vec{b} = \hat{i} - \hat{j} - 3\hat{k}

,

\vec{c} = \vec{a} \times \vec{b}

and

\vec{d} = \vec{c} \times \vec{a}

. Then

\left(\vec{a} - \vec{b}\right) \cdot \vec{d}

is equal to :

(A)4
(B)-4
(C)-2
(D)2

Q18·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}

,

\vec{b} = 2\hat{i} + \hat{j} - \hat{k}

,

\vec{c} = \lambda\hat{i} + \hat{j} + \hat{k}

and

\vec{v} = \vec{a} \times \vec{b}

. If

\vec{v}.\vec{c} = 11

and the length of the projection of

\vec{b}

on

\vec{c}

is p, then

9p^2

is equal to :

(A)9
(B)6
(C)4
(D)12

Q19·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = 2\hat{i} - \hat{j} + \hat{k}

and

\vec{b} = \lambda\hat{j} + 2\hat{k}

,

\lambda \in Z

be two vectors, Let

\vec{c} = \vec{a} \times \vec{b}

and

\vec{d}

be a vector of magnitude 2 in yz-plane. If

|\vec{c}| = \sqrt{53}

, then the maximum possible value of

\left(\vec{c} \cdot \vec{d}\right)^{2}

is equal to :

(A)26
(B)104
(C)208
(D)52

Q20·MathematicsNumericalJEE Main 2026

Let a vector

\vec{a} = \sqrt{2}\hat{i} - \hat{j} + \lambda\hat{k}

,

\lambda > 0

, make an obtuse angle with the vector

\vec{b} = -\lambda^2\hat{i} + 4\sqrt{2}\hat{j} + 4\sqrt{2}\hat{k}

and an angle θ,

\frac{\pi}{6} < \theta < \frac{\pi}{2}

, with the positive z-axis. If the set of all possible values of λ is (α, β) – {γ}, then α + β + γ is equal to ______.

Q21·MathematicsSingle correctJEE Main 2026

Let

\overrightarrow{AB} = 2\hat{i} + 4\hat{j} - 5k

and

\overrightarrow{AD} = \hat{i} + 2\hat{j} + \lambda k

,

\lambda \in \mathbb{R}

. Let the projection of the vector

\vec{v} = \hat{i} + \hat{j} + \hat{k}

on the diagonal

\overrightarrow{AC}

of the parallelogram ABCD be of length one unit. If α, β, where α > β, be the roots of the equation

\lambda^2 x^2 - 6\lambda x + 5 = 0

, then 2α − β is equal to

(A)1
(B)4
(C)3
(D)6

Q22·MathematicsSingle correctJEE Main 2026

Let

\vec{a} = -\hat{i} + 2\hat{j} + 2\hat{k}

,

\vec{b} = 8\hat{i} + 7\hat{j} - 3\hat{k}

and

\vec{c}

be a vector such that

\vec{a} \times \vec{c} = \vec{b}

. If

\vec{c}.(\hat{i} + \hat{j} + \hat{k}) = 4

, then

|\vec{a} + \vec{c}|^{2}

is equal to :

(A)33
(B)30
(C)35
(D)27

Q23·MathematicsSingle correctJEE Main 2026

For a triangle ABC, let

\vec{p} = \overrightarrow{BC}

,

\vec{q} = \overrightarrow{CA}

and

\vec{r} = \overrightarrow{BA}

. If

|\vec{p}| = 2\sqrt{3}

,

|\vec{q}| = 2

and

\cos\theta = \frac{1}{\sqrt{3}}

, where

\theta

is the angle between

\vec{P}

and

\vec{q}

, then

\left|\vec{p} \times \left(\vec{q} - 3\vec{r}\right)\right|^{2} + 3\left|\vec{r}\right|^{2}

is equal to:

(A)340
(B)220
(C)410
(D)200

Q24·MathematicsNumericalJEE Advanced 2025

Consider the vectors

\vec{x} = \hat{i} + 2\hat{j} + 3\hat{k}, \quad \vec{y} = 2\hat{i} + 3\hat{j} + \hat{k},

and

\vec{z} = 3\hat{i} + \hat{j} + 2\hat{k}

. For two distinct positive real numbers α and β, define

\vec{X} = \alpha\vec{x} + \beta\vec{y} - \vec{z}, \quad \vec{Y} = \alpha\vec{y} + \beta\vec{z} - \vec{x},

and

\vec{Z} = \alpha\vec{z} + \beta\vec{x} - \vec{y}

. If the vectors

\vec{X}, \vec{Y}

, and

\vec{Z}

lie in a plane, then the value of α + β − 3 is ______

Q25·MathematicsSingle correctJEE Advanced 2025

Let

\vec{w} = \hat{i} + \hat{j} - 2\hat{k}

, and

\vec{u}

and

\vec{v}

be two vectors, such that

\vec{u} \times \vec{v} = \vec{w}

and

\vec{v} \times \vec{w} = \vec{u}

. Let α, β, γ, and t be real numbers such that

\vec{u} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}

,

-t\alpha + \beta + \gamma = 0

,

\alpha - t\beta + \gamma = 0

, and

\alpha + \beta - t\gamma = 0

. Match each entry in List-I to the correct entries in List-II and choose the correct option. The correct option is:

List-I	List-II
P. $\|\vec{v}\|^2$ is equal to	1.0
Q.If $\alpha = \sqrt{3}$ , then $\gamma^2$ is equal to	2.1
R.If $\alpha = \sqrt{3}$ , then $(\beta + \gamma)^2$ is equal to	3.2
S.If $\alpha = \sqrt{2}$ , then $t + 3$ is equal to	4.3
	5.5

(A)(P) → (2), (Q) → (1), (R) → (4), (S) → (5)
(B)(P) → (2), (Q) → (4), (R) → (3), (S) → (5)
(C)(P) → (2), (Q) → (1), (R) → (4), (S) → (3)
(D)(P) → (5), (Q) → (4), (R) → (1), (S) → (3)

Q26·MathematicsNumericalJEE Advanced 2025

For any two points M and N in the XY –plane, let

\overrightarrow{MN}

denote the vector from M to N, and

\vec{0}

denote the zero vector. Let P, Q and R be three distinct points in the XY-plane. Let S be a point inside the triangle ΔPQR such that

\overrightarrow{SP} + 5\overrightarrow{SQ} + 6\overrightarrow{SR} = \vec{0}

. Let E and F be the mid-points of the sides PR and QR, respectively. Then the value of

\frac{\text{length of the line segment EF}}{\text{length of the line segment ES}}

is __________.

Q27·MathematicsSingle correctJEE Main 2025

Let

\vec a=\hat i+2\hat j+\hat k

and

\vec b=2\hat i+\hat j-\hat k

. Let

\hat c

be a unit vector in the plane of the vectors

\vec a

and

\vec b

and be perpendicular to

\vec a

. Then such a vector

\hat c

is:

(A) $\dfrac{1}{\sqrt2}(\hat j-2\hat k)$
(B) $\dfrac{1}{\sqrt3}(-\hat i-\hat j-\hat k)$
(C) $\dfrac{1}{\sqrt3}(\hat i-\hat j+\hat k)$
(D) $\dfrac{1}{\sqrt2}(-\hat i+\hat k)$

Q28·MathematicsSingle correctJEE Main 2025

Let

\vec{a}

and

\vec{b}

be the vectors of the same magnitude such that

\dfrac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1

. Then

\dfrac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}

is:

(A) $2+4\sqrt{2}$
(B) $1+\sqrt{2}$
(C) $2+\sqrt{2}$
(D) $4+2\sqrt{2}$

Q29·MathematicsSingle correctJEE Main 2025

Let the angle

\theta

,

0<\theta<\dfrac{\pi}{2}

between two unit vectors

\hat{a}

and

\hat{b}

be

\sin^{-1}\left(\dfrac{\sqrt{65}}{9}\right)

. If the vector

\vec{c}=3\hat{a}+6\hat{b}+9(\hat{a}\times\hat{b})

, then the value of

9(\vec{c}\cdot\hat{a})-3(\vec{c}\cdot\hat{b})

is

(A)31
(B)27
(C)29
(D)24

Q30·MathematicsIntegerJEE Main 2025

Let the three sides of a triangle ABC be given by the vectors

2\hat{i}-\hat{j}+\hat{k}

,

\hat{i}-3\hat{j}-5\hat{k}

and

3\hat{i}-4\hat{j}-4\hat{k}

. Let G be the centroid of the triangle ABC. Then

6\left(|AG|^2+|BG|^2+|CG|^2\right)

is equal to ______.

Q31·MathematicsSingle correctJEE Main 2025

Consider two vectors

\vec{u}=3\hat{i}-\hat{j}

and

\vec{v}=2\hat{i}+\hat{j}-\lambda\hat{k}

,

\lambda>0

. The angle between them is given by

\cos^{-1}\left(\dfrac{\sqrt{5}}{2\sqrt{7}}\right)

. Let

\vec{v}=\vec{v_1}+\vec{v_2}

, where

\vec{v_1}

is parallel to

\vec{u}

and

\vec{v_2}

is perpendicular to

\vec{u}

. Then the value of

|\vec{v_1}|^2+|\vec{v_2}|^2

is equal to:

(A) $\dfrac{23}{2}$
(B)14
(C) $\dfrac{25}{2}$
(D)10

Q32·MathematicsIntegerJEE Main 2025

Let

\vec a=\hat i+\hat j+\hat k

,

\vec b=3\hat i+2\hat j-\hat k

,

\vec c=\lambda\hat j+\mu\hat k

and

\hat d

be a unit vector such that

\vec a\times\hat d=\vec b\times\hat d

and

\vec c\cdot\hat d=1

. If

\vec c

is perpendicular to

\vec a

, then

\left|3\lambda\hat d+\mu\vec c\right|^2

is equal to __________.

Q33·MathematicsIntegerJEE Main 2025

Let

\vec a=\hat i+2\hat j+\hat k

,

\vec b=3\hat i-3\hat j+3\hat k

,

\vec c=2\hat i-\hat j+2\hat k

and

\vec d

be a vector such that

\vec b\times\vec d=\vec c\times\vec d

and

\vec a\cdot\vec d=4

. Then

\left|\vec a\times\vec d\right|^2

is equal to __________.

Q34·MathematicsSingle correctJEE Main 2025

If

\vec{a}

is nonzero vector such that its projections on the vectors

2\hat{i}-\hat{j}+2\hat{k}

,

\hat{i}+2\hat{j}-2\hat{k}

and

\hat{k}

are equal, then a unit vector along

\vec{a}

is:

(A) $\dfrac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}+5\hat{k})$
(B) $\dfrac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}-5\hat{k})$
(C) $\dfrac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}+5\hat{k})$
(D) $\dfrac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}-5\hat{k})$

Q35·MathematicsSingle correctJEE Main 2025

Let

\vec{a}=2\hat{i}-3\hat{j}+\hat{k}

,

\vec{b}=3\hat{i}+2\hat{j}+5\hat{k}

and a vector

\vec{c}

be such that

(\vec{a}-\vec{c})\times\vec{b}=-18\hat{i}-3\hat{j}+12\hat{k}

and

\vec{a}\cdot\vec{c}=3

. If

\vec{b}\times\vec{c}=\vec{d}

, then

|\vec{a}\cdot\vec{d}|

is equal to:

(A)18
(B)12
(C)9
(D)15

Q36·MathematicsSingle correctJEE Main 2025

Let

ABCD

be a tetrahedron such that the edges

AB

,

AC

and

AD

are mutually perpendicular. Let the areas of the triangles

ABC

,

ACD

and

ADB

be 5, 6 and 7 square units respectively. Then the area (in square units) of the

\triangle BCD

is equal to:

(A) $\sqrt{340}$
(B)12
(C) $\sqrt{110}$
(D) $7\sqrt{3}$

Q37·MathematicsSingle correctJEE Main 2025

Let

\vec{a}=2\hat{i}-\hat{j}+3\hat{k}

,

\vec{b}=3\hat{i}-5\hat{j}+\hat{k}

and

\vec{c}

be a vector such that

\vec{a}\times\vec{c}=\vec{c}\times\vec{b}

and

(\vec{a}+\vec{c})\cdot(\vec{b}+\vec{c})=168

. Then the maximum value of

|\vec{c}|^2

is:

(A)77
(B)462
(C)308
(D)154

Q38·MathematicsSingle correctJEE Main 2025

Let

\hat{a}

be a unit vector perpendicular to the vectors

\vec{b}=\hat{i}-2\hat{j}+3\hat{k}

and

\vec{c}=2\hat{i}+3\hat{j}-\hat{k}

, and makes an angle of

\cos^{-1}\left(-\dfrac{1}{3}\right)

with the vector

\hat{i}+\hat{j}+\hat{k}

. If

\hat{a}

makes an angle of

\dfrac{\pi}{3}

with the vector

\hat{i}+\alpha\hat{j}+\hat{k}

, then the value of

\alpha

is:

(A) $-\sqrt{3}$
(B) $\sqrt{6}$
(C) $-\sqrt{6}$
(D) $\sqrt{3}$

Q39·MathematicsSingle correctJEE Main 2025

Let

\vec{a}=\hat{i}+2\hat{j}+\hat{k}

and

\vec{b}=2\hat{i}+7\hat{j}+3\hat{k}

. Let

L_1:\vec{r}=(-\hat{i}+2\hat{j}+\hat{k})+\lambda\vec{a}

,

\lambda\in R

and

L_2:\vec{r}=(\hat{j}+\hat{k})+\mu\vec{b}

,

\mu\in R

be two lines. If the line

L_3

passes through the point of intersection of

L_1

and

L_2

, and is parallel to

\vec{a}+\vec{b}

, then

L_3

passes through the point:

(A) $(8, 26, 12)$
(B) $(2, 8, 5)$
(C) $(-1, -1, 1)$
(D) $(5, 17, 4)$

Q40·MathematicsSingle correctJEE Main 2025

If the components of

\vec{a}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}

along and perpendicular to

\vec{b}=3\hat{i}+\hat{j}-\hat{k}

respectively, are

\frac{16}{11}(3\hat{i}+\hat{j}-\hat{k})

and

\frac{1}{11}(-4\hat{i}-5\hat{j}-17\hat{k})

, then

\alpha^2+\beta^2+\gamma^2

is equal to:

(A)23
(B)18
(C)16
(D)26

Q41·MathematicsIntegerJEE Main 2025

Let

\vec{a}=\hat{i}+\hat{j}+\hat{k}

,

\vec{b}=2\hat{i}+2\hat{j}+\hat{k}

and

\vec{d}=\vec{a}\times\vec{b}

. If

\vec{c}

is a vector such that

\vec{a}\cdot\vec{c}=|\vec{c}|

,

|\vec{c}-2\vec{d}|=8

and the angle between

\vec{d}

and

\vec{c}

is

\frac{\pi}{4}

, then

|10-3\vec{b}\cdot\vec{c}|+|\vec{d}\times\vec{c}|^2

is equal to ____.

Q42·MathematicsSingle correctJEE Main 2025

Let A, B, C be three points in xy-plane, whose position vector are given by

\sqrt{3}\hat{i}+\hat{j}

,

\hat{i}+\sqrt{3}\hat{j}

and

a\hat{i}+(1-a)\hat{j}

respectively with respect to the origin O. If the distance of the point C from the point bisecting the angle between the vectors

\vec{OA}

and

\vec{OB}

is

\frac{9}{\sqrt{2}}

, then the sum of all the possible values of a is:

(A)1
(B) $\frac{9}{2}$
(C)0
(D)2

Q43·MathematicsSingle correctJEE Main 2025

Let

\vec a=\hat i+2\hat j+3\hat k

,

\vec b=3\hat i+\hat j-\hat k

and

\vec c

be three vectors such that

\vec c

is coplanar with

\vec a

and

\vec b

. If the vector

\vec c

is perpendicular to

\vec b

and

\vec a\cdot\vec c=5

, then

|\vec c|

is equal to

(A) $\dfrac{1}{3\sqrt2}$
(B) $18$
(C) $16$
(D) $\sqrt{\dfrac{11}{6}}$

Q44·MathematicsSingle correctJEE Main 2025

Let the position vectors of three vertices of a triangle be

4\vec p+\vec q-3\vec r

,

-5\vec p+\vec q+2\vec r

and

2\vec p-\vec q+2\vec r

. If the position vectors of the orthocenter and the circumcenter of the triangle are

\dfrac{\vec p+\vec q+\vec r}{4}

and

\alpha\vec p+\beta\vec q+\gamma\vec r

respectively, then

\alpha+2\beta+5\gamma

is equal to:

(A)3
(B)1
(C)6
(D)4

Q45·MathematicsSingle correctJEE Main 2025

Let

\vec a=3\hat i-\hat j+2\hat k

,

\vec b=\vec a\times(\hat i-3\hat k)

and

\vec c=\vec b\times\hat k

. Then the projection of

\vec c-2\hat j

on

\vec a

is:

(A) $3\sqrt7$
(B) $\sqrt{14}$
(C) $2\sqrt{14}$
(D) $2\sqrt7$

Q46·MathematicsSingle correctJEE Main 2025

Let the point

A

divide the line segment joining the points

P(-1,-1,2)

and

Q(5,5,10)

internally in the ratio

r:1

(r>0)

. If

O

is the origin and

\left(\overrightarrow{OQ}\cdot\overrightarrow{OA}-\tfrac{1}{5}\,|\overrightarrow{OP}\times\overrightarrow{OA}|^2\right)=10

, then the value of

r

is :

(A)14
(B)3
(C) $\sqrt{7}$
(D)7

Q47·MathematicsSingle correctJEE Main 2025

Let the position vectors of the vertices

A

,

B

and

C

of a tetrahedron

ABCD

be

\hat i+2\hat j+\hat k

,

\hat i+3\hat j-2\hat k

and

2\hat i+\hat j-\hat k

respectively. The altitude from the vertex

D

to the opposite face

ABC

meets the median line segment through

A

of the triangle

ABC

at the point

E

. If the length of

AD

is

\dfrac{\sqrt{110}}{3}

and the volume of the tetrahedron is

\dfrac{\sqrt{805}}{6\sqrt2}

, then the position vector of

E

is

(A) $\dfrac{1}{2}(\hat i+4\hat j+7\hat k)$
(B) $\dfrac{1}{12}(7\hat i+4\hat j+3\hat k)$
(C) $\dfrac{1}{6}(12\hat i+12\hat j+\hat k)$
(D) $\dfrac{1}{6}(7\hat i+12\hat j+\hat k)$

Q48·MathematicsSingle correctJEE Main 2025

Let the arc

AC

of a circle subtend a right angle at the centre

O

. If the point

B

on the arc

AC

divides the arc

AC

such that

\dfrac{\text{length of arc }AB}{\text{length of arc }BC}=\dfrac{1}{5}

, and

\overrightarrow{OC}=\alpha\,\overrightarrow{OA}+\beta\,\overrightarrow{OB}

, then

\alpha+\sqrt2(\sqrt3-1)\beta

is equal to

(A) $2-\sqrt3$
(B) $2\sqrt3$
(C) $5\sqrt3$
(D) $2+\sqrt3$

Q49·MathematicsSingle correctJEE Main 2025

Let

\vec{a}

and

\vec{b}

be two unit vectors such that the angle between them is

\dfrac{\pi}{3}

. If

\lambda\vec{a}+2\vec{b}

and

3\vec{a}-\lambda\vec{b}

are perpendicular to each other, then the number of values of

\lambda

in

[-1,3]

is:

(A)3
(B)2
(C)1
(D)0

Q50·MathematicsIntegerJEE Main 2025

Let

\hat c

be the projection vector of

\vec b=\lambda\hat i+4\hat k\ (\lambda>0)

on

\vec a=\hat i+2\hat j+2\hat k

. If

|\vec a+\hat c|=7

, find the area of the parallelogram formed by

\vec b

and

\hat c

.

Q51·MathematicsIntegerJEE Advanced 2024

Let

\vec{p} = 2\hat{i} + \hat{j} + 3\hat{k}

and

\vec{q} = \hat{i} - \hat{j} + \hat{k}

. If for some real numbers

\alpha

,

\beta

, and

\gamma

, we have

15\hat{i} + 10\hat{j} + 6\hat{k} = \alpha\left(2\vec{p} + \vec{q}\right) + \beta\left(\vec{p} - 2\vec{q}\right) + \gamma\left(\vec{p} \times \vec{q}\right)

, then the value of

\gamma

is ______

Q52·MathematicsIntegerJEE Advanced 2024

Let

\overrightarrow{OP} = \frac{\alpha - 1}{\alpha}\hat{i} + \hat{j} + \hat{k}, \overrightarrow{OQ} = \hat{i} + \frac{\beta - 1}{\beta}\hat{j} + \hat{k}

and

\overrightarrow{OR} = \hat{i} + \hat{j} + \frac{1}{2}\hat{k}

be three vectors, where

\alpha, \beta \in R - \{0\}

and

O

denotes the origin. If

\left(\overrightarrow{OP} \times \overrightarrow{OQ}\right) \cdot \overrightarrow{OR} = 0

and the point

(\alpha, \beta, 2)

lies on the plane

3x + 3y - z + l = 0

, then the value of

l

is ______ .

Q53·MathematicsSingle correctJEE Main 2024

Let three vectors

\vec{a} = \alpha\hat{i} + 4\hat{j} + 2\hat{k}

,

\vec{b} = 5\hat{i} + 3\hat{j} + 4\hat{k}

,

\vec{c} = x\hat{i} + y\hat{j} + z\hat{k}

form a triangle such that

\vec{c} = \vec{a} - \vec{b}

and the area of the triangle is

5\sqrt{6}

. If

\alpha

is a positive real number, then

|\vec{c}|^2

is :

(A)16
(B)14
(C)12
(D)10

Q54·MathematicsSingle correctJEE Main 2024

Between the following two statements: Statement-I: Let

\vec{a}=\hat{i}+2\hat{j}-3\hat{k}

and

\vec{b}=2\hat{i}+\hat{j}-\hat{k}

. Then the vector

\vec{r}

satisfying

\vec{a}\times\vec{r}=\vec{a}\times\vec{b}

and

\vec{a}\cdot\vec{r}=0

is of magnitude

\sqrt{10}

. Statement-II: In a triangle ABC,

\cos 2A+\cos 2B+\cos 2C\ge -\dfrac{3}{2}

. In the light of the above statements, choose the correct answer:

(A)Both Statement-I and Statement-II are incorrect
(B)Statement-I is incorrect but Statement-II is correct
(C)Both Statement-I and Statement-II are correct
(D)Statement-I is correct but Statement-II is incorrect

Q55·MathematicsSingle correctJEE Main 2024

Let

\vec{OA} = 2\vec{a}

,

\vec{OB} = 6\vec{a} + 5\vec{b}

and

\vec{OC} = 3\vec{b}

, where O is the origin. If the area of the parallelogram with adjacent sides

\vec{OA}

and

\vec{OC}

is 15 sq. units, then the area (in sq. units) of the quadrilateral OABC is equal to:

(A)38
(B)40
(C)32
(D)35

Q56·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}

,

\vec{b}=-\hat{i}+\hat{k}

,

\vec{c}=\beta\hat{j}-\hat{k}

, where

\alpha

and

\beta

are integers and

\alpha\beta=-6

. Let the values of the ordered pair

(\alpha, \beta)

for which the area of the parallelogram with diagonals

\vec{a}+\vec{b}

and

\vec{b}+\vec{c}

is

\dfrac{\sqrt{21}}{2}

, be

(\alpha_{1}, \beta_{1})

and

(\alpha_{2}, \beta_{2})

. Then

\alpha_{1}^{2}+\beta_{1}^{2}-\alpha_{2}\beta_{2}

is equal to:

(A) $17$
(B) $24$
(C) $21$
(D) $19$

Q57·MathematicsNumericalJEE Main 2024

Let

\vec a=9\hat i-13\hat j+25\hat k

,

\vec b=3\hat i+7\hat j-13\hat k

and

\vec c=17\hat i-2\hat j+\hat k

be three given vectors. If

\vec r

is a vector such that

\vec r\times\vec a=(\vec b+\vec c)\times\vec a

and

\vec r\cdot(\vec b-\vec c)=0

, then

\dfrac{\left|593\vec r+67\vec a\right|^2}{(593)^2}

is equal to ___

Q58·MathematicsSingle correctJEE Main 2024

The set of all

\alpha

, for which the vectors

\vec a=\alpha t\hat i+6\hat j-3\hat k

and

\vec b=t\hat i-2\hat j-2\alpha t\hat k

are inclined at an obtuse angle for all

t\in\mathbb{R}

is:

(A) $[0,1)$
(B) $(-2,0]$
(C) $\left(-\dfrac43,0\right]$
(D) $\left(-\dfrac43,1\right]$

Q59·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=\hat{i}+2\hat{j}+3\hat{k}

,

\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}

and

\vec{c}=3\hat{i}-\hat{j}+\lambda\hat{k}

be three vectors. Let

\vec{r}

be a unit vector along

\vec{b}+\vec{c}

. If

\vec{r}\cdot\vec{a}=3

, then

3\lambda

is equal to

(A) $27$
(B) $25$
(C) $23$
(D) $21$

Q60·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=4\hat{i}-\hat{j}+\hat{k}

,

\vec{b}=11\hat{i}-\hat{j}+\hat{k}

and

\vec{c}

be a vector such that

(\vec{a}+\vec{b})\times\vec{c}=\vec{c}\times(-2\vec{a}+3\vec{b})

. If

(2\vec{a}+3\vec{b})\cdot\vec{c}=1670

, then

|\vec{c}|^2

is equal to

(A) $1627$
(B) $1618$
(C) $1600$
(D) $1609$

Q61·MathematicsSingle correctJEE Main 2024

If

A(3,1,-1)

,

B\left(\tfrac{5}{3},\tfrac{7}{3},\tfrac{1}{3}\right)

,

C(2,2,1)

and

D\left(\tfrac{10}{3},\tfrac{2}{3},-\tfrac{1}{3}\right)

are the vertices of a quadrilateral

ABCD

, then its area is

(A) $\tfrac{4\sqrt{2}}{3}$
(B) $\tfrac{5\sqrt{2}}{3}$
(C) $2\sqrt{2}$
(D) $\tfrac{2\sqrt{2}}{3}$

Q62·MathematicsSingle correctJEE Main 2024

Let

\vec{a} = 6\hat{i} + \hat{j} - \hat{k}

and

\vec{b} = \hat{i} + \hat{j}

. If

\vec{c}

is a vector such that

|\vec{c}| \ge 6

,

\vec{a}\cdot\vec{c} = 6|\vec{c}|

,

|\vec{c} - \vec{a}| = 2\sqrt{2}

and the angle between

\vec{a}\times\vec{b}

and

\vec{c}

is

60^\circ

, then

|(\vec{a}\times\vec{b})\times\vec{c}|

is equal to:

(A) $\tfrac{9}{2}(6 - \sqrt{6})$
(B) $\tfrac{3}{2}\sqrt{3}$
(C) $\tfrac{3}{2}\sqrt{6}$
(D) $\tfrac{9}{2}(6 + \sqrt{6})$

Q63·MathematicsNumericalJEE Main 2024

Let

\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}

,

\vec{b}=3\hat{i}+4\hat{j}-5\hat{k}

, and a vector

\vec{c}

be such that

\vec{a}\times(\vec{b}+\vec{c})+\vec{b}\times\vec{c}=\hat{i}+8\hat{j}+13\hat{k}

. If

\vec{a}\cdot\vec{c}=13

, then

24-\vec{b}\cdot\vec{c}

is equal to _______.

Q64·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=2\hat{i}+\hat{j}-\hat{k}

,

\vec{b}=\left(\left(\vec{a}\times(\hat{i}+\hat{j})\right)\times\hat{i}\right)\times\hat{i}

. Then the square of the projection of

\vec{a}

on

\vec{b}

is:

(A) $\frac{1}{5}$
(B) $2$
(C) $\frac{1}{3}$
(D) $\frac{2}{3}$

Q65·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=2\hat{i}+5\hat{j}-\hat{k}

,

\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}

and

\vec{c}

be three vectors such that

(\vec{c}+\hat{i})\times(\vec{a}+\vec{b}+\hat{i})=\vec{a}\times(\vec{c}+\hat{i})

and

\vec{a}\cdot\vec{c}=-29

. Then

\vec{c}\cdot(-2\hat{i}+\hat{j}+\hat{k})

is equal to:

(A)10
(B)5
(C)15
(D)12

Q66·MathematicsNumericalJEE Main 2024

Let

\vec{a} = \hat{i} - 3\hat{j} + 7\hat{k}

,

\vec{b} = 2\hat{i} - \hat{j} + \hat{k}

and

\vec{c}

be a vector such that

(\vec{a} + 2\vec{b}) \times \vec{c} = 3(\vec{c} \times \vec{a})

. If

\vec{a} \cdot \vec{c} = 130

, then

\vec{b} \cdot \vec{c}

is equal to ___.

Q67·MathematicsSingle correctJEE Main 2024

Consider three vectors

\vec{a},\vec{b},\vec{c}

. Let

|\vec{a}|=2

,

|\vec{b}|=3

and

\vec{a}=\vec{b}\times\vec{c}

. If

\alpha\in\left[0,\dfrac{\pi}{3}\right]

is the angle between the vectors

\vec{b}

and

\vec{c}

, then the minimum value of

27|\vec{c}-\vec{a}|^2

is equal to:

(A)110
(B)105
(C)124
(D)121

Q68·MathematicsSingle correctJEE Main 2024

If

A(1, -1, 2)

,

B(5, 7, -6)

,

C(3, 4, -10)

and

D(-1, -4, -2)

are the vertices of a quadrilateral ABCD, then its area is:

(A) $12\sqrt{29}$
(B) $24\sqrt{29}$
(C) $24\sqrt{7}$
(D) $48\sqrt{7}$

Q69·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=\hat{i}+\hat{j}+\hat{k}

,

\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}

and

\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}

,

x\in\mathbb{R}

. If

\vec{d}

is the unit vector in the direction of

\vec{b}+\vec{c}

such that

\vec{a}\cdot\vec{d}=1

, then

(\vec{a}\times\vec{b})\cdot\vec{c}

is equal to

(A)9
(B)6
(C)3
(D)11

Q70·MathematicsSingle correctJEE Main 2024

Let a unit vector which makes an angle of

60^\circ

with

2\hat i+2\hat j-\hat k

and an angle of

45^\circ

with

\hat i-\hat k

be

\vec C

. Then

\vec C+\left(-\dfrac12\hat i+\dfrac{1}{3\sqrt2}\hat j-\dfrac{\sqrt2}{3}\hat k\right)

is:

(A) $-\dfrac{\sqrt2}{3}\hat i+\dfrac{\sqrt2}{3}\hat j+\left(\dfrac12+\dfrac{2\sqrt2}{3}\right)\hat k$
(B) $\dfrac{\sqrt2}{3}\hat i+\dfrac{1}{3\sqrt2}\hat j-\dfrac12\hat k$
(C) $\left(\dfrac{1}{\sqrt3}+\dfrac12\right)\hat i+\left(\dfrac{1}{\sqrt3}-\dfrac{1}{3\sqrt2}\right)\hat j+\left(\dfrac{1}{\sqrt3}+\dfrac{\sqrt2}{3}\right)\hat k$
(D) $\dfrac{\sqrt2}{3}\hat i-\dfrac12\hat k$

Q71·MathematicsSingle correctJEE Main 2024

If

\lambda>0

, let

\theta

be the angle between the vectors

\vec{a}=\hat{i}+\lambda\hat{j}-3\hat{k}

and

\vec{b}=3\hat{i}-\hat{j}+2\hat{k}

. If the vectors

\vec{a}+\vec{b}

and

\vec{a}-\vec{b}

are mutually perpendicular, then the value of

(14\cos\theta)^2

is equal to

(A)25
(B)20
(C)50
(D)40

Q72·MathematicsNumericalJEE Main 2024

Let

ABC

be a triangle of area

15\sqrt2

and the vectors

\vec{AB}=\hat i+2\hat j-7\hat k

,

\vec{BC}=a\hat i+b\hat j+c\hat k

and

\vec{AC}=6\hat i+d\hat j-2\hat k

,

d>0

. Then the square of the length of the largest side of the triangle

ABC

is ___

Q73·MathematicsNumericalJEE Main 2024

Let

\vec a=\hat i+\hat j+\hat k

,

\vec b=-\hat i-8\hat j+2\hat k

and

\vec c=4\hat i+c_2\hat j+c_3\hat k

be three vectors such that

\vec b\times\vec a=\vec c\times\vec a

. If the angle between the vector

\vec c

and the vector

3\hat i+4\hat j+\hat k

is

\theta

, then the greatest integer less than or equal to

\tan^2\theta

is __________.

Q74·MathematicsSingle correctJEE Main 2024

Consider a

\triangle ABC

where

A(1,2,3)

,

B(-2,8,0)

and

C(3,6,7)

. If the angle bisector of

\angle BAC

meets the line BC at D, then the length of the projection of the vector

\vec{AD}

on the vector

\vec{AC}

is:

(A) $\dfrac{37}{2\sqrt{38}}$
(B) $\dfrac{\sqrt{38}}{2}$
(C) $\dfrac{39}{2\sqrt{38}}$
(D) $\sqrt{19}$

Q75·MathematicsSingle correctJEE Main 2024

Let

\vec a=-5\hat i+\hat j-3\hat k

,

\vec b=\hat i+2\hat j-4\hat k

and

\vec c=\big(\big(\big((\vec a\times\vec b)\times\hat i\big)\times\hat i\big)\times\hat i\big)

. Then

\vec c\cdot(-\hat i+\hat j+\hat k)

is equal to:

(A) $-12$
(B) $-10$
(C) $-13$
(D) $-15$

Q76·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=3\hat{i}+\hat{j}-2\hat{k}

,

\vec{b}=4\hat{i}+\hat{j}+7\hat{k}

and

\vec{c}=\hat{i}-3\hat{j}+4\hat{k}

be three vectors. If a vector

\vec{p}

satisfies

\vec{p}\times\vec{b}=\vec{c}\times\vec{b}

and

\vec{p}\cdot\vec{a}=0

, then

\vec{p}\cdot\left(\hat{i}-\hat{j}-\hat{k}\right)

is equal to

(A) $24$
(B) $36$
(C) $28$
(D) $32$

Q77·MathematicsNumericalJEE Main 2024

Let

\vec{a}

and

\vec{b}

be two vectors such that

|\vec{a}|=1

,

|\vec{b}|=4

and

\vec{a}\cdot\vec{b}=2

. If

\vec{c}=(2\vec{a}\times\vec{b})-3\vec{b}

and the angle between

\vec{b}

and

\vec{c}

is

\alpha

, then

192\sin^2\alpha

is equal to ______.

Q78·MathematicsNumericalJEE Main 2024

Let

\vec{a}=3\hat{i}+2\hat{j}+\hat{k}

,

\vec{b}=2\hat{i}-\hat{j}+3\hat{k}

and

\vec{c}

be a vector such that

(\vec{a}+\vec{b})\times\vec{c}=2(\vec{a}\times\vec{b})+24\hat{j}-6\hat{k}

and

(\vec{a}-\vec{b}+\hat{i})\cdot\vec{c}=-3

. Then

|\vec{c}|^2

is equal to ______.

Q79·MathematicsSingle correctJEE Main 2024

Let

\vec{a}=\hat{i}+\alpha\hat{j}+\beta\hat{k}

,

\alpha,\beta\in R

. Let a vector

\vec{b}

be such that the angle between

\vec{a}

and

\vec{b}

is

\dfrac{\pi}{4}

and

|\vec{b}|^2=6

. If

\vec{a}\cdot\vec{b}=3\sqrt{2}

, then the value of

(\alpha^2+\beta^2)\,|\vec{a}\times\vec{b}|^2

is equal to:

(A) $90$
(B) $75$
(C) $95$
(D) $85$

Q80·MathematicsSingle correctJEE Main 2024

Let

A(2,3,5)

and

C(-3,4,-2)

be opposite vertices of a parallelogram

ABCD

. If the diagonal

\vec{BD}=\hat i+2\hat j+3\hat k

, then the area of the parallelogram is equal to:

(A) $\dfrac12\sqrt{410}$
(B) $\dfrac12\sqrt{474}$
(C) $\dfrac12\sqrt{586}$
(D) $\dfrac12\sqrt{306}$

Q81·MathematicsSingle correctJEE Main 2024

Let

\vec{a}

and

\vec{b}

be two vectors such that

|\vec{b}|=1

and

|\vec{b}\times\vec{a}|=2

. Then

\left|(\vec{b}\times\vec{a})-\vec{b}\right|^2

is equal to:

(A) $3$
(B) $5$
(C) $1$
(D) $4$

Q82·MathematicsSingle correctJEE Main 2024

Let

\vec a=a_1\hat i+a_2\hat j+a_3\hat k

and

\vec b=b_1\hat i+b_2\hat j+b_3\hat k

be two vectors such that

|\vec a|=1

,

\vec a\cdot\vec b=2

and

|\vec b|=4

. If

\vec c=2(\vec a\times\vec b)-3\vec b

, then the angle between

\vec b

and

\vec c

is equal to:

(A) $\cos^{-1}\left(\dfrac{2}{\sqrt3}\right)$
(B) $\cos^{-1}\left(-\dfrac{1}{\sqrt3}\right)$
(C) $\cos^{-1}\left(-\dfrac{\sqrt3}{2}\right)$
(D) $\cos^{-1}\left(\dfrac{2}{3}\right)$

Q83·MathematicsSingle correctJEE Main 2024

Let a unit vector

\hat u=x\hat i+y\hat j+z\hat k

make angles

\dfrac{\pi}{2},\dfrac{\pi}{3}

and

\dfrac{2\pi}{3}

with the vectors

\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat j

,

\dfrac{1}{\sqrt2}\hat j+\dfrac{1}{\sqrt2}\hat k

and

\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat k

respectively. If

\vec v=\dfrac{1}{\sqrt2}\hat i+\dfrac{1}{\sqrt2}\hat j+\dfrac{1}{\sqrt2}\hat k

, then

|\hat u-\vec v|^2

is equal to:

(A) $\dfrac{11}{2}$
(B) $\dfrac{5}{2}$
(C)9
(D)7

Q84·MathematicsSingle correctJEE Main 2024

Let O be the origin and the position vector of A and B be

2\hat{i}+2\hat{j}+\hat{k}

and

2\hat{i}+4\hat{j}+4\hat{k}

respectively. If the internal bisector of

\angle AOB

meets the line AB at C, then the length of OC is

(A) $\dfrac{2}{3}\sqrt{31}$
(B) $\dfrac{2}{3}\sqrt{34}$
(C) $\dfrac{3}{4}\sqrt{34}$
(D) $\dfrac{3}{2}\sqrt{31}$

Q85·MathematicsSingle correctJEE Main 2024

Let

\overrightarrow{OA}=\vec a,\ \overrightarrow{OB}=12\vec a+4\vec b

and

\overrightarrow{OC}=\vec b

, where

O

is the origin. If

S

is the parallelogram with adjacent sides

OA

and

OC

, then

\dfrac{\text{area of the quadrilateral }OABC}{\text{area of }S}

is equal to:

(A)6
(B)10
(C)7
(D)8

Q86·MathematicsSingle correctJEE Main 2024

Let

\vec{a},\vec{b}

and

\vec{c}

be three non-zero vectors such that

\vec{b}

and

\vec{c}

are non-collinear. If

\vec{a}+5\vec{b}

is collinear with

\vec{c}

,

\vec{b}+6\vec{c}

is collinear with

\vec{a}

and

\vec{a}+\alpha\vec{b}+\beta\vec{c}=\vec{0}

, then

\alpha+\beta

is equal to

(A) $35$
(B) $30$
(C) $-30$
(D) $-25$

Q87·MathematicsSingle correctJEE Main 2024

Let the position vectors of the vertices A, B and C of a triangle be

2\hat i+2\hat j+\hat k

,

\hat i+2\hat j+2\hat k

and

2\hat i+\hat j+2\hat k

respectively. Let

l_1

,

l_2

and

l_3

be the lengths of perpendiculars drawn from the ortho center of the triangle on the sides AB, BC and CA respectively, then

l_1^2+l_2^2+l_3^2

equals :

(A) $\dfrac15$
(B) $\dfrac12$
(C) $\dfrac14$
(D) $\dfrac13$

Q88·MathematicsSingle correctJEE Main 2024

Let

\vec a=\hat i+2\hat j+\hat k

,

\vec b=3(\hat i-\hat j+\hat k)

. Let

\vec c

be the vector such that

\vec a\times\vec c=\vec b

and

\vec a\cdot\vec c=3

. Then

\vec a\cdot\big((\vec c\times\vec b)-\vec b-\vec c\big)

is equal to:

(A)32
(B)24
(C)20
(D)36

Q89·MathematicsSingle correctJEE Main 2024

The position vectors of the vertices A, B and C of a triangle are

2\hat i-3\hat j+3\hat k

,

2\hat i+2\hat j+3\hat k

and

-\hat i+\hat j+3\hat k

respectively. Let

l

denotes the length of the angle bisector AD of

\angle BAC

where D is on the line segment BC, then

2l^2

equals :

(A)49
(B)42
(C)50
(D)45

Q90·MathematicsNumericalJEE Main 2024

The least positive integral value of

\alpha

, for which the angle between the vectors

\alpha\hat i-2\hat j+2\hat k

and

\alpha\hat i+2\alpha\hat j-2\hat k

is acute, is __________.

Q91·MathematicsNumericalJEE Advanced 2023

Let P be the plane

\sqrt{3}x + 2y + 3z = 16

and let

S = \left\{\alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane P is } \frac{7}{2}\right\}

. Let

\vec{u}, \vec{v}

and

\vec{w}

be three distinct vectors in S such that

\left|\vec{u} - \vec{v}\right| = \left|\vec{v} - \vec{w}\right| = \left|\vec{w} - \vec{u}\right|

. Let V be the volume of the parallelepiped determined by vectors,

\vec{u}, \vec{v}

and

\vec{w}

. Then the value of

\frac{80}{\sqrt{3}}V

is

Q92·MathematicsSingle correctJEE Advanced 2023

Let the position vectors of the points

P

,

Q

,

R

and S be

\vec{a} = \hat{i} + 2\hat{j} - 5\hat{k}

,

\vec{b} = 3\hat{i} + 6\hat{j} + 3\hat{k}

,

\vec{c} = \frac{17}{5}\hat{i} + \frac{16}{5}\hat{j} + 7\hat{k}

and

\vec{d} = 2\hat{i} + \hat{j} + \hat{k}

, respectively. Then which of the following statements is true?

(A)The points $P$ , $Q$ , $R$ and S are NOT coplanar
(B) $\frac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ internally in the ratio $5 : 4$
(C) $\frac{\vec{b} + 2\vec{d}}{3}$ is the position vector of a point which divides $PR$ externally in the ratio $5 : 4$
(D)The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is 95

Q93·MathematicsSingle correctJEE Main 2023

Let

S

be the set of all

(\lambda, \mu)

for which the vectors

\lambda\hat{i} - \hat{j} + \hat{k}

,

\hat{i} + 2\hat{j} + \mu\hat{k}

and

3\hat{i} - 4\hat{j} + 5\hat{k}

, where

\lambda - \mu = 5

, are coplanar, then

\sum_{(\lambda,\mu)\in S} 80(\lambda^2 + \mu^2)

is equal to:

(A) $2370$
(B) $2130$
(C) $2290$
(D) $2210$

Q94·MathematicsSingle correctJEE Main 2023

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and

\left(\vec{AB}-\vec{BC}\right)+\left(\vec{AD}-\vec{DC}\right)=k\,\vec{FE}

, then k is equal to

(A)2
(B) $-2$
(C) $-4$
(D)4

Q95·MathematicsSingle correctJEE Main 2023

Let

|\vec{a}| = 2

,

|\vec{b}| = 3

and the angle between the vectors

\vec{a}

and

\vec{b}

be

\dfrac{\pi}{4}

. Then

\left|(\vec{a} + 2\vec{b}) \times (2\vec{a} - 3\vec{b})\right|^2

is equal to

(A)482
(B)441
(C)841
(D)882

Q96·MathematicsSingle correctJEE Main 2023

Let

\vec a=\hat i+4\hat j+2\hat k,\,\vec b=3\hat i-2\hat j+7\hat k

and

\vec c=2\hat i-\hat j+4\hat k

. If a vector

\vec d

satisfies

\vec d\times\vec b=\vec c\times\vec b

and

\vec a\cdot\vec d=24

, then

|\vec d|^{2}

is equal to:

(A) $413$
(B) $423$
(C) $323$
(D) $313$

Q97·MathematicsSingle correctJEE Main 2023

Let for a triangle ABC,

\overrightarrow{AB} = -2\hat{i} + \hat{j} + 3\hat{k}

,

\overrightarrow{CB} = \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k}

,

\overrightarrow{CA} = 4\hat{i} + 3\hat{j} + \delta\hat{k}

. If

\delta > 0

and the area of the triangle ABC is

5\sqrt{6}

, then

\overrightarrow{CB} \cdot \overrightarrow{CA}

is equal to

(A)60
(B)120
(C)108
(D)54

Q98·MathematicsNumericalJEE Main 2023

Let

\vec a=3\hat i+\hat j-\hat k

and

\vec c=2\hat i-3\hat j+3\hat k

. If

\vec b

is a vector such that

\vec a=\vec b\times\vec c

and

|\vec b|=\sqrt{50}

, then

\left|72-|\vec b+\vec c|^{2}\right|

is equal to _____.

Q99·MathematicsSingle correctJEE Main 2023

Let

\lambda\in\mathbb{Z}

,

\vec a=2\lambda\hat i+\hat j-\hat k

and

\vec b=3\hat i-\hat j+2\hat k

. Let

\vec c

be a vector such that

(\vec a+\vec b+\vec c)\times\vec c=\vec 0

,

\vec a\cdot\vec c=-17

and

\vec b\cdot\vec c=-20

. Then

|\vec c\times(\lambda\hat i+\hat j+\hat k)|^2

is equal to

(A)62
(B)46
(C)53
(D)49

Q100·MathematicsSingle correctJEE Main 2023

Let a, b, c be three distinct real numbers, none equal to one. If the vectors

a\hat i+\hat j+\hat k

,

\hat i+b\hat j+\hat k

and

\hat i+\hat j+c\hat k

are coplanar, then

\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}

is equal to

(A)1
(B)-1
(C)3
(D)-2

Q101·MathematicsSingle correctJEE Main 2023

If four distinct points with position vectors

\vec{a},\vec{b},\vec{c}

and

\vec{d}

are coplanar, then

\left[\vec{a}\,\vec{b}\,\vec{c}\right]

is equal to

(A) $\left[\vec{d}\,\vec{c}\,\vec{a}\right]+\left[\vec{b}\,\vec{a}\,\vec{d}\right]+\left[\vec{c}\,\vec{d}\,\vec{b}\right]$
(B) $\left[\vec{d}\,\vec{b}\,\vec{a}\right]+\left[\vec{a}\,\vec{c}\,\vec{d}\right]+\left[\vec{d}\,\vec{b}\,\vec{c}\right]$
(C) $\left[\vec{a}\,\vec{d}\,\vec{b}\right]+\left[\vec{d}\,\vec{c}\,\vec{a}\right]+\left[\vec{b}\,\vec{d}\,\vec{c}\right]$
(D) $\left[\vec{b}\,\vec{c}\,\vec{d}\right]+\left[\vec{a}\,\vec{b}\,\vec{c}\right]+\left[\vec{d}\,\vec{b}\,\vec{a}\right]$

Q102·MathematicsSingle correctJEE Main 2023

For any vector

\vec{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}

, with

|a_i| < 1

,

i = 1, 2, 3

, consider the following statements: (A):

\max \{ |a_1|, |a_2|, |a_3| \} \le |\vec{a}|

(B):

|\vec{a}| \le 3 \max \{ |a_1|, |a_2|, |a_3| \}

(A)Only (B) is true
(B)Only (A) is true
(C)Neither (A) nor (B) is true
(D)Both (A) and (B) are true

Q103·MathematicsNumericalJEE Main 2023

Let

\vec{a}=\hat{i}+2\hat{j}+3\hat{k}

and

\vec{b}=\hat{i}+\hat{j}-\hat{k}

. If

\vec{c}

is a vector such that

\vec{a}\cdot\vec{c}=11

,

\vec{b}\cdot(\vec{a}\times\vec{c})=27

and

\vec{b}\cdot\vec{c}=-\sqrt{3}\,|\vec{b}|

, then

|\vec{a}\times\vec{c}|^2

is equal to

Q104·MathematicsSingle correctJEE Main 2023

An arc PQ of a circle subtends a right angle at its centre O. The mid point of the arc PQ is R. If

\overrightarrow{OP}=\vec u

,

\overrightarrow{OR}=\vec v

and

\overrightarrow{OQ}=\alpha\vec u+\beta\vec v

, then

\alpha,\beta^2

are the roots of the equation

(A) $x^2-x-2=0$
(B) $3x^2+2x-1=0$
(C) $x^2+x-2=0$
(D) $3x^2-2x-1=0$

Q105·MathematicsSingle correctJEE Main 2023

If the points P and Q are respectively the circumcentre and the orthocentre of a

\triangle ABC

, then

\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}

is equal to:

(A) $2\overrightarrow{QP}$
(B) $\overrightarrow{QP}$
(C) $2\overrightarrow{PQ}$
(D) $\overrightarrow{PQ}$

Q106·MathematicsSingle correctJEE Main 2023

Let

\vec{a}=2\hat{i}+7\hat{j}-\hat{k}

,

\vec{b}=3\hat{i}+5\hat{k}

and

\vec{c}=\hat{i}-\hat{j}+2\hat{k}

. Let

\vec{d}

be a vector which is perpendicular to both

\vec{a}

and

\vec{b}

, and

\vec{c}\cdot\vec{d}=12

. Then

\left(-\hat{i}+\hat{j}-\hat{k}\right)\cdot\left(\vec{c}\times\vec{d}\right)

is equal to:

(A) $48$
(B) $42$
(C) $44$
(D) $24$

Q107·MathematicsSingle correctJEE Main 2023

Let O be the origin and the position vector of the point P be

-\hat i-2\hat j+3\hat k

. If the position vectors of the points A, B and C are

-2\hat i+\hat j-3\hat k

,

2\hat i+4\hat j-2\hat k

and

-4\hat i+2\hat j-\hat k

respectively, then the projection of the vector

\overrightarrow{OP}

on a vector perpendicular to the vectors

\overrightarrow{AB}

and

\overrightarrow{AC}

is

(A) $3$
(B) $\dfrac83$
(C) $\dfrac{10}{3}$
(D) $\dfrac73$

Q108·MathematicsNumericalJEE Main 2023

Let

\vec{a}=6\hat{i}+9\hat{j}+12\hat{k}

,

\vec{b}=\alpha\hat{i}+11\hat{j}-2\hat{k}

and

\vec{c}

be vectors such that

\vec{a}\times\vec{c}=\vec{a}\times\vec{b}

. If

\vec{a}\cdot\vec{c}=-12

,

\vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5

, then

\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})

is equal to

Q109·MathematicsSingle correctJEE Main 2023

Let the vectors

\vec{u}_1=\hat{i}+\hat{j}+a\hat{k}

,

\vec{u}_2=\hat{i}+b\hat{j}+\hat{k}

and

\vec{u}_3=c\hat{i}+\hat{j}+\hat{k}

be coplanar. If the vectors

\vec{v}_1=(a+b)\hat{i}+c\hat{j}+c\hat{k}

,

\vec{v}_2=a\hat{i}+(b+c)\hat{j}+a\hat{k}

and

\vec{v}_3=b\hat{i}+b\hat{j}+(c+a)\hat{k}

are also coplanar, then

6(a+b+c)

is equal to

(A)0
(B)6
(C)12
(D)4

Q110·MathematicsSingle correctJEE Main 2023

If the points with vectors

\alpha\hat{i}+10\hat{j}+13\hat{k}

,

6\hat{i}+11\hat{j}+11\hat{k}

,

\frac{9}{2}\hat{i}+\beta\hat{j}-8\hat{k}

are collinear, then

(19\alpha-6\beta)^{2}

is equal to

(A)36
(B)16
(C)25
(D)49

Q111·MathematicsSingle correctJEE Main 2023

The area of the quadrilateral ABCD with vertices

A(2,1,1)

,

B(1,2,5)

,

C(-2,-3,5)

and

D(1,-6,-7)

is equal to

(A)48
(B) $8\sqrt{38}$
(C)54
(D) $9\sqrt{38}$

Q112·MathematicsSingle correctJEE Main 2023

Let the vectors

\vec{a},\vec{b},\vec{c}

represent three coterminous edges of a parallelepiped of volume

V

. Then the volume of the parallelepiped, whose coterminous edges are represented by

\vec{a},\ \vec{b}+\vec{c}

and

\vec{a}+2\vec{b}+3\vec{c}

is equal to

(A)3V
(B)6V
(C)2V
(D)V

Q113·MathematicsSingle correctJEE Main 2023

The sum of all values of

\alpha

, for which the points whose position vectors

\hat{i}-2\hat{j}+3\hat{k}

,

2\hat{i}-3\hat{j}+4\hat{k}

,

(\alpha+1)\hat{i}+2\hat{k}

and

9\hat{i}+(\alpha-8)\hat{j}+6\hat{k}

are coplanar, is equal to

(A)6
(B)4
(C) $-2$
(D)2

Q114·MathematicsSingle correctJEE Main 2023

Let

\vec a=2\hat i+3\hat j+4\hat k,\,\vec b=2\hat i-2\hat j-2\hat k

and

\vec c=-\hat i+4\hat j+3\hat k

. If

\vec d

is a vector perpendicular to both

\vec b

and

\vec c

and

\vec a\cdot\vec d=18

, then

|\vec a\times\vec d|^2

is equal to:

(A) $640$
(B) $760$
(C) $680$
(D) $720$

Q115·MathematicsSingle correctJEE Main 2023

Let the position vectors of the points

A,\,B,\,C

and

D

be

5\hat i+5\hat j+2\lambda\hat k,\,\hat i+2\hat j+3\hat k,\,-2\hat i+\lambda\hat j+4\hat k

and

-\hat i+5\hat j+6\hat k

. Let the set

S=\{\lambda\in\mathbb{R}: A,B,C\text{ and }D\text{ are coplanar}\}

. Then

\displaystyle\sum_{\lambda\in S}(\lambda+2)^{2}

is equal to:

(A) $41$
(B) $25$
(C) $13$
(D) $\dfrac{37}{2}$

Q116·MathematicsNumericalJEE Main 2023

Let

\vec{v}=\alpha\hat{i}+2\hat{j}-3\hat{k}

,

\vec{w}=2\alpha\hat{i}+\hat{j}-\hat{k}

and

\vec{u}

be a vector such that

|\vec{u}|=\alpha>0

. If the minimum value of the scalar triple product

[\vec{u}\ \vec{v}\ \vec{w}]

is

-\alpha\sqrt{3401}

, and

|\vec{u}\cdot\hat{i}|^2=\frac{m}{n}

where

m

and

n

are coprime natural numbers, then

m+n

is equal to _______ .

Q117·MathematicsSingle correctJEE Main 2023

Let

\vec a=5\hat i-\hat j-3\hat k

and

\vec b=\hat i+3\hat j+5\hat k

be two vectors. Then which one of the following statements is TRUE?

(A)Projection of $\vec a$ on $\vec b$ is $\dfrac{13}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec b$ .
(B)Projection of $\vec a$ on $\vec b$ is $\dfrac{13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec b$ .
(C)Projection of $\vec a$ on $\vec b$ is $\dfrac{-13}{\sqrt{35}}$ and the direction of the projection vector is same as of $\vec b$ .
(D)Projection of $\vec a$ on $\vec b$ is $\dfrac{-13}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec b$ .

Q118·MathematicsSingle correctJEE Main 2023

Let

\vec a=2\hat i-7\hat j+5\hat k

,

\vec b=\hat i+\hat k

and

\vec c=\hat i+2\hat j-3\hat k

be three given vectors. If

\vec r

is a vector such that

\vec r\times\vec a=\vec c\times\vec a

and

\vec r\cdot\vec b=0

, then

|\vec r|

is equal to:

(A) $\dfrac{11}{5}\sqrt2$
(B) $\dfrac{\sqrt{914}}{7}$
(C) $\dfrac{11}{7}\sqrt2$
(D) $\dfrac{11}{7}$

Q119·MathematicsNumericalJEE Main 2023

A(2,6,2)

,

B(-4,0,\lambda)

,

C(2,3,-1)

and

D(4,5,0)

,

|\lambda|\le 5

are the vertices of a quadrilateral

ABCD

. If its area is

18

square units, then

5-6\lambda

is equal to _______ .

Q120·MathematicsSingle correctJEE Main 2023

Let

\vec a=\hat i+2\hat j+3\hat k,\ \vec b=\hat i-\hat j+2\hat k

and

\vec c=5\hat i-3\hat j+3\hat k

be three vectors. If

\vec r

is a vector such that

\vec r\times\vec b=\vec c\times\vec b

and

\vec r\cdot\vec a=0

, then

25|\vec r|^2

is equal to

(A) $560$
(B) $449$
(C) $339$
(D) $336$

Q121·MathematicsNumericalJEE Main 2023

Let

\vec a,\vec b,\vec c

be three vectors such that

|\vec a|=\sqrt{31},\ 4|\vec b|=|\vec c|=2

and

2(\vec a\times\vec b)=3(\vec c\times\vec a)

. If the angle between

\vec b

and

\vec c

is

\dfrac{2\pi}{3}

, then

\left(\dfrac{\vec a\times\vec c}{\vec a\cdot\vec b}\right)^2

is equal to

Q122·MathematicsSingle correctJEE Main 2023

Let

\vec{a}=2\hat{i}+\hat{j}+\hat{k}

, and

\vec{b}

and

\vec{c}

be two nonzero vectors such that

|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|

and

\vec{b}\cdot\vec{c}=0

. Consider the following two statements: (A)

|\vec{a}+\lambda\vec{c}|\ge|\vec{a}|

for all

\lambda\in\mathbb{R}

. (B)

\vec{a}

and

\vec{c}

are always parallel. Then.

(A)both (A) and (B) are correct
(B)only (A) is correct
(C)neither (A) nor (B) is correct
(D)only (B) is correct

Q123·MathematicsNumericalJEE Main 2023

Let

\vec{a}

and

\vec{b}

be two vectors such that

|\vec{a}| = \sqrt{14}

,

|\vec{b}| = \sqrt{6}

and

|\vec{a} \times \vec{b}| = \sqrt{48}

. Then

(\vec{a} \cdot \vec{b})^2

is equal to

Q124·MathematicsSingle correctJEE Main 2023

Let

\lambda \in \mathbb{R}

,

\vec{a}=\lambda \hat{i} + 2\hat{j} - 3\hat{k}

,

\vec{b}=\hat{i} - \lambda\hat{j} + 2\hat{k}

. If

\left((\vec{a} + \vec{b}) \times (\vec{a} \times \vec{b})\right) \times (\vec{a} - \vec{b})=8\hat{i} - 40\hat{j} - 24\hat{k}

, then

\left|\lambda(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})\right|^2

is equal to:

(A) $132$
(B) $136$
(C) $140$
(D) $144$

Q125·MathematicsSingle correctJEE Main 2023

Let

\vec{a}

and

\vec{b}

be two vectors. Let

|\vec{a}|=1

,

|\vec{b}|=4

and

\vec{a} \cdot \vec{b}=2

. If

\vec{c}=(2\vec{a} \times \vec{b}) - 3\vec{b}

, then the value of

\vec{b} \cdot \vec{c}

is:

(A) $-24$
(B) $-84$
(C) $-48$
(D) $-60$

Q126·MathematicsSingle correctJEE Main 2023

If

\vec a,\vec b,\vec c

are three non-zero vectors and

\hat n

is a unit vector perpendicular to

\vec c

such that

\vec a=\alpha\vec b-\hat n

,

(\alpha\neq0)

and

\vec b\cdot\vec c=12

, then

|\vec c\times(\vec a\times\vec b)|

is equal to:

(A) $9$
(B) $15$
(C) $6$
(D) $12$

Q127·MathematicsSingle correctJEE Main 2023

If

\vec a=\hat i+2\hat k,\,\vec b=\hat i+\hat j+\hat k,\,\vec c=7\hat i-3\hat j+4\hat k,\,\vec r\times\vec b+\vec b\times\vec c=\vec 0

and

\vec r\cdot\vec a=0

. Then

\vec r\cdot\vec c

is equal to:

(A) $32$
(B) $30$
(C) $36$
(D) $34$

Q128·MathematicsSingle correctJEE Main 2023

If the vectors

\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}

,

\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}

and

\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}

are coplanar and the projection of

\vec{a}

on the vector

\vec{b}

is

\sqrt{54}

units, then the sum of all possible values of

\lambda + \mu

is equal to

(A)0
(B)24
(C)6
(D)18

Q129·MathematicsNumericalJEE Main 2023

Let

\vec{a}

,

\vec{b}

and

\vec{c}

be three non-zero non-coplanar vectors. Let the position vectors of four points

A

,

B

,

C

and

D

be

\vec{a} - \vec{b} + \vec{c}

,

\lambda\vec{a} - 3\vec{b} + 4\vec{c}

,

-\vec{a} + 2\vec{b} - 3\vec{c}

and

2\vec{a} - 4\vec{b} + 6\vec{c}

respectively. If

\vec{AB}

,

\vec{AC}

and

\vec{AD}

are coplanar, then

\lambda

is equal to ________ .

Q130·MathematicsSingle correctJEE Main 2023

Let

\vec a=4\hat i+3\hat j

and

\vec b=3\hat i-4\hat j+5\hat k

. If

\vec c

is a vector such that

\vec c\cdot(\vec a\times\vec b)+25=0,\,\vec c\cdot(\hat i+\hat j+\hat k)=4

, and projection of

\vec c

on

\vec a

is

1

, then the projection of

\vec c

on

\vec b

equals:

(A) $\dfrac15$
(B) $\dfrac{5}{\sqrt2}$
(C) $\dfrac{3}{\sqrt2}$
(D) $\dfrac{1}{\sqrt2}$

Q131·MathematicsSingle correctJEE Main 2023

Let

\vec{a},\vec{b}

and

\vec{c}

be three non zero vectors such that

\vec{b}\cdot\vec{c}=0

and

\vec{a}\times(\vec{b}\times\vec{c})=\dfrac{\vec{b}-\vec{c}}{2}

. If

\vec{d}

be a vector such that

\vec{b}\cdot\vec{d}=\vec{a}\cdot\vec{b}

, then

(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})

is equal to:

(A) $-\dfrac{1}{4}$
(B) $\dfrac{1}{4}$
(C) $\dfrac{3}{4}$
(D) $\dfrac{1}{2}$

Q132·MathematicsSingle correctJEE Main 2023

Let

\vec{a} = -\hat{i} - \hat{j} + \hat{k}

,

\vec{a} \cdot \vec{b} = 1

and

\vec{a} \times \vec{b} = \hat{i} - \hat{j}

. Then

\vec{a} - 6\vec{b}

is equal to

(A) $3(\hat{i} - \hat{j} + \hat{k})$
(B) $(\hat{i} + \hat{j} - \hat{k})$
(C) $3(\hat{i} + \hat{j} + \hat{k})$
(D) $3(\hat{i} - \hat{j} - \hat{k})$

Q133·MathematicsSingle correctJEE Main 2023

The vector

\vec{a}=-\hat{i}+2\hat{j}+\hat{k}

is rotated through a right angle, passing through the

y

-axis in its way and the resulting vector is

\vec{b}

. Then the projection of

3\vec{a}+\sqrt{2}\,\vec{b}

on

\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}

is:

(A) $2\sqrt{3}$
(B) $1$
(C) $3\sqrt{2}$
(D) $\sqrt{6}$

Q134·MathematicsSingle correctJEE Main 2023

If the four points, whose position vectors are

3\hat{i} - 4\hat{j} + 2\hat{k}

,

\hat{i} + 2\hat{j} - \hat{k}

,

-2\hat{i} - \hat{j} + 3\hat{k}

and

5\hat{i} - 2\alpha\hat{j} + 4\hat{k}

are coplanar, then

\alpha

is equal to

(A) $\dfrac{73}{17}$
(B) $\dfrac{107}{17}$
(C) $\dfrac{-73}{17}$
(D) $\dfrac{-107}{17}$

Q135·MathematicsSingle correctJEE Main 2023

Let

PQR

be a triangle. The points

A

,

B

and

C

are on the sides

QR

,

RP

and

PQ

respectively such that

\dfrac{QA}{AR}=\dfrac{RB}{BP}=\dfrac{PC}{CQ}=\dfrac{1}{2}

. Then

\dfrac{\text{Area}(\triangle PQR)}{\text{Area}(\triangle ABC)}

is equal to

(A) $4$
(B) $3$
(C) $\tfrac{5}{2}$
(D) $2$

Q136·MathematicsSingle correctJEE Main 2023

Let

\vec\alpha=4\hat i+3\hat j+5\hat k

and

\vec\beta=\hat i+2\hat j-4\hat k

. Let

\vec{\beta_1}

be parallel to

\vec\alpha

and

\vec{\beta_2}

be perpendicular to

\vec\alpha

. If

\vec\beta=\vec{\beta_1}+\vec{\beta_2}

, then the value of

5\vec{\beta_2}\cdot(\hat i+\hat j+\hat k)

is

(A) $7$
(B) $9$
(C) $6$
(D) $11$

Q137·MathematicsSingle correctJEE Main 2023

Let

\vec{u}=\hat{i}-\hat{j}-2\hat{k}

,

\vec{v}=2\hat{i}+\hat{j}-\hat{k}

,

\vec{v}\cdot\vec{w}=2

and

\vec{v}\times\vec{w}=\vec{u}+\lambda\vec{v}

. Then

\vec{u}\cdot\vec{w}

is equal to

(A) $2$
(B) $\tfrac{3}{2}$
(C) $1$
(D) $-\tfrac{2}{3}$

Q138·MathematicsNumericalJEE Main 2023

Let

\vec a=\hat i+2\hat j+\lambda\hat k,\ \vec b=3\hat i-5\hat j-\lambda\hat k,\ \vec a\cdot\vec c=7,\ 2\vec b\cdot\vec c+43=0,\ \vec a\times\vec c=\vec b\times\vec c

. Then

|\vec a\cdot\vec b|

is equal to

Q139·MathematicsMultiple correctJEE Advanced 2022

Let

\hat{i}

,

\hat{j}

and

\hat{k}

be the unit vectors along the three positive coordinate axes. Let

\vec{a} = 3\hat{i} + \hat{j} - \hat{k}

,

\vec{b} = \hat{i} + b_{2}\hat{j} + b_{3}\hat{k}

, b

_{2}

, b

_{3}

∈ R

\vec{c} = c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}

, c

_{1}

, c

_{2}

, c

_{3}

∈ R be the vectors such that b

_{2}

b

_{3}

> 0,

\vec{a} \cdot \vec{b} = 0

and

\begin{pmatrix} 0 & -c_{3} & c_{2} \\ c_{3} & 0 & -c_{1} \\ -c_{2} & c_{1} & 0 \end{pmatrix}\begin{pmatrix} 1 \\ b_{2} \\ b_{3} \end{pmatrix} = \begin{pmatrix} 3-c_{1} \\ 1-c_{2} \\ -1-c_{3} \end{pmatrix}

Then, which of the following is/are TRUE?

(A) $\vec{a} \cdot \vec{c} = 0$
(B) $\vec{b} \cdot \vec{c} = 0$
(C) $\left|\vec{b}\right| > \sqrt{10}$
(D) $\left|\vec{c}\right| \leq \sqrt{11}$ .

Q140·MathematicsNumericalJEE Main 2022

Let

\vec{a}

and

\vec{b}

be two vectors such that

|\vec{a} + \vec{b}|^2 = |\vec{a}|^2 + 2|\vec{b}|^2

,

\vec{a} \cdot \vec{b} = 3

and

|\vec{a} \times \vec{b}|^2 = 75

. Then

|\vec{a}|^2

is equal to____.

Q141·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = 3\hat{i} + \hat{j}

and

\vec{b} = \hat{i} + 2\hat{j} + \hat{k}

. Let

\vec{c}

be a vector satisfying

\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda\vec{c}

. If

\vec{b}

and

\vec{c}

are non-parallel, then the value of

\lambda

is :

(A) $-5$
(B)5
(C)1
(D) $-1$

Q142·MathematicsSingle correctJEE Main 2022

Let

\hat{a}

and

\hat{b}

be two unit vectors such that the angle between them is

\frac{\pi}{4}

. If

\theta

is the angle between the vectors

(\hat{a} + \hat{b})

and

(\hat{a} + 2\hat{b} + 2(\hat{a} \times \hat{b}))

, then the value of

164 \cos^{2}\theta

is equal to :

(A) $90 + 27\sqrt{2}$
(B) $45 + 18\sqrt{2}$
(C) $90 + 3\sqrt{2}$
(D) $54 + 90\sqrt{2}$

Q143·MathematicsSingle correctJEE Main 2022

Let

\vec{a}, \vec{b}, \vec{c}

be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and

\left(\vec{a} \times \vec{b}\right) \cdot \left(\vec{b} \times \vec{c}\right) + \left(\vec{b} \times \vec{c}\right) \cdot \left(\vec{c} \times \vec{a}\right) + \left(\vec{c} \times \vec{a}\right) \cdot \left(\vec{a} \times \vec{b}\right) = 168

then

|\vec{a}| + |\vec{b}| + |\vec{c}|

is equal to:

(A)10
(B)14
(C)16
(D)18

Q144·MathematicsSingle correctJEE Main 2022

Let the vectors

\vec{a} = (1+t)\hat{i} + (1-t)\hat{j} + \hat{k}

,

\vec{b} = (1-t)\hat{i} + (1+t)\hat{j} + 2\hat{k}

and

\vec{c} = t\hat{i} - t\hat{j} + \hat{k}

,

t \in R

be such that for

\alpha, \beta, \gamma \in R

,

\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c} = \vec{0} \Rightarrow \alpha = \beta = \gamma = 0

. Then, the set of all values of t is :

(A)a non-empty finite set
(B)equal to N
(C)equal to R − {0}
(D)equal to R

Q145·MathematicsSingle correctJEE Main 2022

Let S be the set of all

a \in R

for which the angle between the vectors

\vec{u}=a\left(\log_{e}b\right)\hat{i}-6\hat{j}+3\hat{k}

and

\vec{v}=\left(\log_{e}b\right)\hat{i}+2\hat{j}+2a\left(\log_{e}b\right)\hat{k}, (b>1)

is acute. Then S is equal to

(A) $\left(-\infty,-\frac{4}{3}\right)$
(B) $\Phi$
(C) $\left(-\frac{4}{3},0\right)$
(D) $\left(\frac{12}{7},\infty\right)$

Q146·MathematicsSingle correctJEE Main 2022

Let a vector

\vec{a}

has a magnitude 9. Let a vector

\vec{b}

be such that for every

(x, y) \in R \times R - \{(0,0)\}

, the vector

(x\vec{a} + y\vec{b})

is perpendicular to the vector

(6y\,\vec{a} - 18x\,\vec{b})

. Then the value of

\left|\vec{a} \times \vec{b}\right|

is equal to:

(A) $9\sqrt{3}$
(B) $27\sqrt{3}$
(C)9
(D)81

Q147·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = \alpha\hat{i} + \hat{j} + \beta\hat{k}

and

\vec{b} = 3\hat{i} - 5\hat{j} + 4\hat{k}

be two vectors, such that

\vec{a} \times \vec{b} = -\hat{i} + 9\hat{i} + 12k

. Then the projection of

\vec{b} - 2\vec{a}

on

\vec{b} + \vec{a}

is equal to

(A)2
(B) $\frac{39}{5}$
(C)9
(D) $\frac{46}{5}$

Q148·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = 2\hat{i} - \hat{j} + 5\hat{k}

and

\vec{b} = \alpha\hat{i} + \beta\hat{j} + 2\hat{k}

. If

\left(\left(\vec{a} \times \vec{b}\right) \times \hat{i}\right).\hat{k} = \frac{23}{2}

, then

\left| \vec{b} \times 2\hat{j} \right|

is equal to

(A)4
(B)5
(C) $\sqrt{21}$
(D) $\sqrt{17}$

Q149·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = \alpha\hat{i} + \hat{j} - \hat{k}

and

\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}, \alpha > 0

. If the projection of

\vec{a} \times \vec{b}

on the vector

-\hat{i} + 2\hat{j} - 2\hat{k}

is 30, then α is equal to

(A) $\dfrac{15}{2}$
(B)8
(C) $\dfrac{13}{2}$
(D)7

Q150·MathematicsSingle correctJEE Main 2022

A vector

\vec{a}

is parallel to the line of intersection of the plane determined by the vectors

\hat{i},\hat{i}+\hat{j}

and the plane determined by the vectors

\hat{i}-\hat{j},\hat{i}+\hat{k}

. The obtuse angle between

\vec{a}

and the vector

\vec{b}=\hat{i}-2\hat{j}+2\hat{k}

is

(A) $\frac{3\pi}{4}$
(B) $\frac{2\pi}{3}$
(C) $\frac{4\pi}{5}$
(D) $\frac{5\pi}{6}$

Q151·MathematicsSingle correctJEE Main 2022

Let ABC be a triangle such that

\overrightarrow{BC} = \vec{a}

,

\overrightarrow{CA} = \vec{b}

,

\overrightarrow{AB} = \vec{c}

,

|\vec{a}| = 6\sqrt{2}

,

|\vec{b}| = 2\sqrt{3}

and

\vec{b} \cdot \vec{c} = 12

. Consider the statements : (S1) :

|(\vec{a} \times \vec{b}) + (\vec{c} \times \vec{b})| - |\vec{c}| = 6(2\sqrt{2} - 1)

(S2) :

\angle ABC = \cos^{-1}\left( \sqrt{\frac{2}{3}} \right)

. Then

(A)both (S1) and (S2) are true
(B)only (S1) is true
(C)only (S2) is true
(D)both (S1) and (S2) are false

Q152·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = \alpha\hat{i} + 3\hat{j} - \hat{k}, \vec{b} = 3\hat{i} - \beta\hat{j} + 4\hat{k}

and

\vec{c} = \hat{i} + 2\hat{j} - 2\hat{k}

where α, β ∈ R, be three vectors. If the projection of

\vec{a}

on

\vec{c}

is

\frac{10}{3}

and

\vec{b} \times \vec{c} = -6\hat{i} + 10\hat{j} + 7\hat{k}

, then the value of α + β equal to :

(A)3
(B)4
(C)5
(D)6

Q153·MathematicsSingle correctJEE Main 2022

Let A, B, C be three points whose position vectors respectively are:

\vec{a} = \hat{i} + 4\hat{j} + 3\hat{k}

\vec{b} = 2\hat{i} + \alpha\hat{j} + 4\hat{k}, \alpha \in \mathbb{R}

\vec{c} = 3\hat{i} - 2\hat{j} + 5\hat{k}

If

\alpha

is the smallest positive integer for which

\vec{a}, \vec{b}, \vec{c}

are non-collinear, then the length of the median, in

\triangle ABC

, through A is:

(A) $\frac{\sqrt{82}}{2}$
(B) $\frac{\sqrt{62}}{2}$
(C) $\frac{\sqrt{69}}{2}$
(D) $\frac{\sqrt{66}}{2}$

Q154·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = \alpha\hat{i} + 2\hat{j} - \hat{k}

and

\vec{b} = -2\hat{i} + \alpha\hat{j} + \hat{k}

, where

\alpha \in \mathbb{R}

. If the area of the parallelogram whose adjacent sides are represented by the vectors

\vec{a}

and

\vec{b}

is

\sqrt{15\left(\alpha^{2} + 4\right)}

, then the value of

2\left|\vec{a}\right|^{2} + \left(\vec{a} \cdot \vec{b}\right)\left|\vec{b}\right|^{2}

is equal to

(A)10
(B)7
(C)9
(D)14

Q155·MathematicsSingle correctJEE Main 2022

Let

\vec{a}

be a vector which is perpendicular to the vector

3\hat{i} + \frac{1}{2}\hat{j} + 2\hat{k}

. If

\vec{a} \times \left(2\hat{i} + \hat{k}\right) = 2\hat{i} - 13\hat{j} - 4\hat{k}

, then the projection of the vector

\vec{a}

on the vector

2\hat{i} + 2\hat{j} + \hat{k}

is

(A) $\frac{1}{3}$
(B) $1$
(C) $\frac{5}{3}$
(D) $\frac{7}{3}$

Q156·MathematicsNumericalJEE Main 2022

If

\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k},

\vec{b} = 3\hat{i} + 3\hat{j} + \hat{k}

and

\vec{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}

are coplanar vectors and

\vec{a} \cdot \vec{c} = 5,

\vec{b} \perp \vec{c},

then

122\,(c_1 + c_2 + c_3)

is equal to ______.

Q157·MathematicsSingle correctJEE Main 2022

Let

\vec{a}

and

\vec{b}

be the vectors along the diagonal of a parallelogram having area

2\sqrt{2}

. Let the angle between

\vec{a}

and

\vec{b}

be acute.

|\vec{a}|=1

and

|\vec{a}\cdot\vec{b}|=|\vec{a}\times\vec{b}|

. If

\vec{c}=2\sqrt{2}\left(\vec{a}\times\vec{b}\right)-2\vec{b}

, then an angle between

\vec{b}

and

\vec{c}

is :

(A) $\dfrac{\pi}{4}$
(B) $-\dfrac{\pi}{4}$
(C) $\dfrac{5\pi}{6}$
(D) $\dfrac{3\pi}{4}$

Q158·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = \hat{i} + \hat{j} - \hat{k}

and

\vec{c} = 2\hat{i} - 3\hat{j} + 2\hat{k}

. Then the number of vectors

\vec{b}

such that

\vec{b} \times \vec{c} = \vec{a}

and

\left|\vec{b}\right| \in \{1, 2, ....., 10\}

is :

(A)0
(B)1
(C)2
(D)3

Q159·MathematicsSingle correctJEE Main 2022

If

\vec{a}\cdot\vec{b}=1

,

\vec{b}\cdot\vec{c}=2

and

\vec{c}\cdot\vec{a}=3

, then the value of

\left[\vec{a}\times\left(\vec{b}\times\vec{c}\right),\ \vec{b}\times\left(\vec{c}\times\vec{a}\right),\ \vec{c}\times\left(\vec{b}\times\vec{a}\right)\right]

is :

(A) $0$
(B) $-6\vec{a}\cdot\left(\vec{b}\times\vec{c}\right)$
(C) $12\vec{c}\cdot\left(\vec{a}\times\vec{b}\right)$
(D) $-12\vec{b}\cdot\left(\vec{c}\times\vec{a}\right)$

Q160·MathematicsSingle correctJEE Main 2022

Let

\vec{a}=\hat{i}+\hat{j}+2\hat{k}

,

\vec{b}=2\hat{i}-3\hat{j}+\hat{k}

and

\vec{c}=\hat{i}-\hat{j}+\hat{k}

be three given vectors. Let

\vec{v}

be a vector in the plane of

\vec{a}

and

\vec{b}

whose projection on

\vec{c}

is

\frac{2}{\sqrt{3}}

. If

\vec{v}\cdot\hat{j}=7

, then

\vec{v}\cdot\left(\hat{i}+\hat{k}\right)

is equal to :

(A)6
(B)7
(C)8
(D)9

Q161·MathematicsNumericalJEE Main 2022

Let θ be the angle between the vectors

\vec{a}

and

\vec{b}

, where

|\vec{a}| = 4

,

|\vec{b}| = 3

\theta \in \left(\frac{\pi}{4}, \frac{\pi}{3}\right)

. Then

\left|\left(\vec{a} - \vec{b}\right) \times \left(\vec{a} + \vec{b}\right)\right|^2 + 4\left(\vec{a}\cdot\vec{b}\right)^2

is equal to ____

Q162·MathematicsNumericalJEE Main 2022

Let

\vec{b} = \hat{i} + \hat{j} + \lambda\hat{k}, \lambda \in R

. If

\vec{a}

is a vector such that

\vec{a} \times \vec{b} = 13\hat{i} - \hat{j} - 4\hat{k}

and

\vec{a} \cdot \vec{b} + 21 = 0

, then

\left(\vec{b} - \vec{a}\right) \cdot \left(\hat{k} - \hat{j}\right) + \left(\vec{b} + \vec{a}\right) \cdot \left(\hat{i} - \hat{k}\right)

is equal to

Q163·MathematicsSingle correctJEE Main 2022

Let

\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}

a_i > 0

, i = 1, 2, 3 be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of

\vec{a}

on the vector

3\hat{i} + 4\hat{j}

be 7. Let

\vec{b}

be a vector obtained by rotating

\vec{a}

with 90°. If

\vec{a}

,

\vec{b}

and x-axis are coplanar, then projection of a vector

\vec{b}

on

3\hat{i} + 4\hat{j}

is equal to

(A) $\sqrt{7}$
(B) $\sqrt{2}$
(C) $2$
(D) $7$

Q164·MathematicsSingle correctJEE Main 2022

Let

\hat{a},\hat{b}

be unit vectors. If

\vec{c}

be a vector such that the angle between

\hat{a}

and

\vec{c}

is

\dfrac{\pi}{12}

, and

\hat{b}=\vec{c}+2(\vec{c}\times\hat{a})

, then

\left|6\vec{c}\right|^{2}

is equal to

(A) $6(3-\sqrt{3})$
(B) $3+\sqrt{3}$
(C) $6(3+\sqrt{3})$
(D) $6(\sqrt{3}+1)$

Q165·MathematicsSingle correctJEE Main 2022

Let

\hat{a}

and

\hat{b}

be two unit vectors such that

\left|\left(\hat{a}+\hat{b}\right)+2\left(\hat{a}\times\hat{b}\right)\right|=2

. If

\theta \in (0, \pi)

is the angle between

\hat{a}

and

\hat{b}

, then among the statements : (S1) :

2\left|\hat{a}\times\hat{b}\right|=\left|\hat{a}-\hat{b}\right|

(S2) : The projection of

\hat{a}

on

\left(\hat{a}+\hat{b}\right)

is

\dfrac{1}{2}

(A)Only (S1) is true
(B)Only (S2) is true
(C)Both (S1) and (S2) are true
(D)Both (S1) and (S2) are false

Q166·MathematicsIntegerJEE Advanced 2021

Let

\vec{u}, \vec{v}

and

\vec{w}

be vectors in three-dimensional space, where

\vec{u}

and

\vec{v}

are unit vectors which are not perpendicular to each other and

\vec{u}\cdot\vec{w} = 1

,

\vec{v}\cdot\vec{w} = 1

,

\vec{w}\cdot\vec{w} = 4

If the volume of the parallelopiped, whose adjacent sides are represented by the vectors

\vec{u}, \vec{v}

and

\vec{w}

, is

\sqrt{2}

, then the value of

\left|3\vec{u} + 5\vec{v}\right|

is____.

Q167·MathematicsMultiple correctJEE Advanced 2021

Let O be the origin and

\vec{OA} = 2\hat{i} + 2\hat{j} + \hat{k}

,

\vec{OB} = \hat{i} - 2\hat{j} + 2\hat{k}

and

\vec{OC} = \frac{1}{2}\left(\vec{OB} - \lambda\vec{OA}\right)

for some λ > 0. If

\left|\vec{OB} \times \vec{OC}\right| = \frac{9}{2}

, then which of the following statements is (are) TRUE?

(A)Projection of $\vec{OC}$ on $\vec{OA}$ is $-\frac{3}{2}$
(B)Area of the triangle OAB is $\frac{9}{2}$
(C)Area of the triangle ABC is $\frac{9}{2}$
(D)The acute angle between the diagonals of the parallelogram with adjacent sides $\vec{OA}$ and $\vec{OC}$ is $\frac{\pi}{3}$

Q168·MathematicsNumericalJEE Main 2021

Let

\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k}

and

\vec{b} = \hat{i} + 2\hat{j} - \hat{k}

. Let a vector

\vec{v}

be in the plane containing

\vec{a}

and

\vec{b}

. If

\vec{v}

is perpendicular to the vector

3\hat{i} + 2\hat{j} - \hat{k}

and its projection on

\vec{a}

is 19 units, then

\left|2\vec{v}\right|^{2}

is equal to ______ .

Q169·MathematicsSingle correctJEE Main 2021

Let

\vec{a}

and

\vec{b}

be two vectors such that

\left|2\vec{a} + 3\vec{b}\right| = \left|3\vec{a} + \vec{b}\right|

and the angle between

\vec{a}

and

\vec{b}

is

60^{\circ}

. If

\frac{1}{8}\vec{a}

is a unit vector, then

\left|\vec{b}\right|

is equal to :

(A)4
(B)6
(C)5
(D)8

Q170·MathematicsSingle correctJEE Main 2021

Let

\vec{a}, \vec{b}, \vec{c}

be three vectors mutually perpendicular to each other and have same magnitude. If a vector

\vec{r}

satisfies.

\vec{a} \times \{(\vec{r} - \vec{b}) \times \vec{a}\} + \vec{b} \times \{(\vec{r} - \vec{c}) \times \vec{b}\} + \vec{c} \times \{(\vec{r} - \vec{a}) \times \vec{c}\} = \vec{0}

, then

\vec{r}

is equal to :

(A) $\frac{1}{3}(\vec{a} + \vec{b} + \vec{c})$
(B) $\frac{1}{3}(2\vec{a} + \vec{b} - \vec{c})$
(C) $\frac{1}{2}(\vec{a} + \vec{b} + \vec{c})$
(D) $\frac{1}{2}(\vec{a} + \vec{b} + 2\vec{c})$

Q171·MathematicsNumericalJEE Main 2021

Let

\vec{a} = \hat{i} + 5\hat{j} + \alpha\hat{k}

,

\vec{b} = \hat{i} + 3\hat{j} + \beta\hat{k}

and

\vec{c} = -\hat{i} + 2\hat{j} - 3\hat{k}

be three vectors such that,

\left|\vec{b} \times \vec{c}\right| = 5\sqrt{3}

and

\vec{a}

is perpendicular to

\vec{b}

. Then the greatest amongst the values of

\left|\vec{a}\right|^{2}

is _______.

Q172·MathematicsNumericalJEE Main 2021

If the projection of the vector

\hat{i} + 2\hat{j} + \hat{k}

on the sum of the two vectors

2\hat{i} + 4\hat{j} - 5\hat{k}

and

-\lambda\hat{i} + 2\hat{j} + 3\hat{k}

is 1, then

\lambda

is equal to ________.

Q173·MathematicsSingle correctJEE Main 2021

Let

\vec{a} = \hat{i} + \hat{j} + \hat{k}

and

\vec{b} = \hat{j} - \hat{k}

. If

\vec{c}

is a vector such that

\vec{a} \times \vec{c} = \vec{b}

and

\vec{a} \cdot \vec{c} = 3

, then

\vec{a} \cdot (\vec{b} \times \vec{c})

is equal to :

(A) $-2$
(B) $-6$
(C) $6$
(D) $2$

Q174·MathematicsNumericalJEE Main 2021

Let

\vec{a}=\hat{i}+\hat{j}+\hat{k}

,

\vec{b}

and

\vec{c}=\hat{j}-\hat{k}

be three vectors such that

\vec{a} \times \vec{b}=\vec{c}

and

\vec{a} \cdot \vec{b}=1

. If the length of projection vector of the vector

\vec{b}

on the vector

\vec{a} \times \vec{c}

is

\ell

, then the value of

3\ell^2

is equal to ____.

Q175·MathematicsSingle correctJEE Main 2021

Let

\vec{a}, \vec{b}

and

\vec{c}

be three vectors such that

\vec{a} = \vec{b} \times \left( \vec{b} \times \vec{c} \right)

. If magnitudes of the vectors

\vec{a}, \vec{b}

and

\vec{c}

are √2, 1 and 2 respectively and the angle between

\vec{b}

and

\vec{c}

is

\theta\left( 0 < \theta < \frac{\pi}{2} \right)

, then the value of 1 + tan θ is equal to :

(A)√3 + 1
(B) $\frac{\sqrt{3} + 1}{\sqrt{3}}$
(C)2
(D)1

Q176·MathematicsNumericalJEE Main 2021

Let

\vec{a}

= î − αĵ + βk̂,

\vec{b}

= 3î + βĵ − αk̂ and

\vec{c}

= −αî − 2ĵ + k̂, where α and β are integers. If

\vec{a}\cdot\vec{b}

= −1 and

\vec{b}\cdot\vec{c}

= 10, then

\left(\vec{a}\times\vec{b}\right)\cdot\vec{c}

is equal to…….

Q177·MathematicsSingle correctJEE Main 2021

Let

\vec{a} = \hat{i} + \hat{j} + 2\hat{k}

and

\vec{b} = -\hat{i} + 2\hat{j} + 3\hat{k}

. Then the vector product

\left(\vec{a} + \vec{b}\right) \times \left(\vec{a} \times \left(\left(\vec{a} - \vec{b}\right) \times \vec{b}\right)\right) \times \vec{b}

is equal to :

(A) $5\left(34\hat{i} - 5\hat{j} + 3\hat{k}\right)$
(B) $5\left(30\hat{i} - 5\hat{j} + 7\hat{k}\right)$
(C) $7\left(30\hat{i} - 5\hat{j} + 7\hat{k}\right)$
(D) $7\left(34\hat{i} - 5\hat{j} + 3\hat{k}\right)$

Q178·MathematicsSingle correctJEE Main 2021

Let the vectors

\left(2+a+b\right)\hat{i}+\left(a+2b+c\right)\hat{j}-\left(b+c\right)\hat{k},

\left(1+b\right)\hat{i}+2b\hat{j}-b\hat{k}

and

\left(2+b\right)\hat{i}+2b\hat{j}+\left(1-b\right)\hat{k}, a,b,c \in R

be co-planar. Then which of the following is true?

(A) $2a=b+c$
(B) $2b=a+c$
(C) $a=b+2c$
(D) $3c=a+b$

Q179·MathematicsSingle correctJEE Main 2021

If

\left|\vec{a}\right| = 2, \left|\vec{b}\right| = 5

and

\left|\vec{a} \times \vec{b}\right| = 8

, then

\left|\vec{a} . \vec{b}\right|

is equal to :

(A)5
(B)4
(C)6
(D)3

Q180·MathematicsSingle correctJEE Main 2021

Let a, b and c be distinct positive numbers. If the vectors

a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}

and

c\hat{i}+c\hat{j}+b\hat{k}

are co-planar, then c is equal to :

(A) $\frac{2}{\frac{1}{a}+\frac{1}{b}}$
(B) $\frac{a+b}{2}$
(C) $\frac{1}{a}+\frac{1}{b}$
(D) $\sqrt{ab}$

Q181·MathematicsNumericalJEE Main 2021

Let

\vec{p} = 2\hat{i} + 3\hat{j} + \hat{k}

and

\vec{q} = \hat{i} + 2\hat{j} + \hat{k}

be two vectors. If a vector :

\vec{r} = \left( \alpha\hat{i} + \beta\hat{j} + \gamma\hat{k} \right)

is perpendicular to each of the vectors

\left( \vec{p} + \vec{q} \right)

and

\left( \vec{p} - \vec{q} \right)

, and

\left| \vec{r} \right| = \sqrt{3}

, then

\left| \alpha \right| + \left| \beta \right| + \left| \gamma \right|

is equal to......

Q182·MathematicsNumericalJEE Main 2021

If

\left(\vec{a} + 3\vec{b}\right)

is perpendicular to

\left(7\vec{a} - 5\vec{b}\right)

and

\left(\vec{a} - 4\vec{b}\right)

is perpendicular to

\left(7\vec{a} - 2\vec{b}\right)

, then the angle between

\vec{a}

and

\vec{b}

(in degrees) is.......

Q183·MathematicsSingle correctJEE Main 2021

Let a vector

\vec{a}

be coplanar with vectors

\vec{b} = 2\hat{i} + \hat{j} + \hat{k}

and

\vec{c} = \hat{i} - \hat{j} + \hat{k}

. If

\vec{a}

is perpendicular to

\vec{d} = 3\hat{i} + 2\hat{j} + 6\hat{k}

and

\left|\vec{a}\right| = \sqrt{10}

. Then a possible value of

\left[\vec{a}\vec{b}\vec{c}\right] + \left[\vec{a}\vec{b}\vec{d}\right] + \left[\vec{a}\vec{c}\vec{d}\right]

is equal to :

(A) $-40$
(B) $-38$
(C) $-42$
(D) $-29$

Q184·MathematicsSingle correctJEE Main 2021

Let three vectors

\vec{a}, \vec{b}

and

\vec{c}

be such that

\vec{a} \times \vec{b} = \vec{c}

,

\vec{b} \times \vec{c} = \vec{a}

and

\left|\vec{a}\right| = 2

. Then which one of the following is not true?

(A)Projection of $\vec{a}$ on $\left(\vec{b} \times \vec{c}\right)$ is 2
(B) $\left[\vec{a}\vec{b}\vec{c}\right] + \left[\vec{c}\vec{a}\vec{b}\right] = 8$
(C) $\left|3\vec{a} + \vec{b} - 2\vec{c}\right|^2 = 51$
(D) $\vec{a} \times \left(\left(\vec{b} + \vec{c}\right) \times \left(\vec{b} - \vec{c}\right)\right) = \vec{0}$

Q185·MathematicsSingle correctJEE Main 2021

Let

\vec{a} = 2\hat{i}+\hat{j}-2\hat{k}

and

\vec{b} = \hat{i}+\hat{j}

. If

\vec{c}

is a vector such that

\vec{a}.\vec{c} = |\vec{c}|

,

\left|\vec{c}-\vec{a}\right| = 2\sqrt{2}

and the angle between

\left(\vec{a} \times \vec{b}\right)

and

\vec{c}

is

\frac{\pi}{6}

, then the value of

\left|\left(\vec{a} \times \vec{b}\right) \times \vec{c}\right|

is :

(A)3
(B) $\frac{2}{3}$
(C) $\frac{3}{2}$
(D)4

Q186·MathematicsNumericalJEE Main 2021

For

p>0

, a vector

\vec{v}_{2}=2\hat{i}+(p+1)\hat{j}

is obtained by rotating the vector

\vec{v}_{1}=\sqrt{3}\,p\hat{i}+\hat{j}

by an angle

\theta

about origin in counter clockwise direction. If

\tan\theta=\frac{\left(\alpha\sqrt{3}-2\right)}{\left(4\sqrt{3}+3\right)}

, then the value of

\alpha

is equal to……….

Q187·MathematicsSingle correctJEE Main 2021

In a triangle ABC, if

\left|\overrightarrow{BC}\right|=3,\left|\overrightarrow{CA}\right|=5

and

\left|\overrightarrow{BA}\right|=7,

then the projection of the vector

\overrightarrow{BA}

on

\overrightarrow{BC}

is equal to :

(A) $\frac{19}{2}$
(B) $\frac{13}{2}$
(C) $\frac{15}{2}$
(D) $\frac{11}{2}$

Q188·MathematicsNumericalJEE Main 2021

Let

\vec{a}, \vec{b}, \vec{c}

be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle

\theta

, with the vector

\vec{a} + \vec{b} + \vec{c}

. Then

36\cos^2 2\theta

is equal to.......

Q189·MathematicsSingle correctJEE Main 2021

In a triangle ABC, if

\left|\vec{BC}\right| = 8

,

\left|\vec{CA}\right| = 7

,

\left|\vec{AB}\right| = 10

, then the projection of the vector

\vec{AB}

on

\vec{AC}

is equal to :

(A) $\frac{25}{4}$
(B) $\frac{85}{14}$
(C) $\frac{127}{20}$
(D) $\frac{115}{16}$

Q190·MathematicsSingle correctJEE Main 2021

A vector

\vec{a}

has components

3p

and

1

with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system,

\vec{a}

has components

p + 1

and

\sqrt{10}

, then a value of p is equal to:

(A) $1$
(B) $-\dfrac{5}{4}$
(C) $\dfrac{4}{5}$
(D) $-1$

Q191·MathematicsSingle correctJEE Main 2021

Let

\vec{a}

and

\vec{b}

be two non-zero vectors perpendicular to each other and

\left|\vec{a}\right| = \left|\vec{b}\right|

. If

\left|\vec{a} \times \vec{b}\right| = \left|\vec{a}\right|

, then the angle between the vectors

\left(\vec{a} + \vec{b} + \left(\vec{a} \times \vec{b}\right)\right)

and

\vec{a}

is equal to :

(A) $\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$
(B) $\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$
(C) $\cos^{-1}\left(\frac{1}{\sqrt{2}}\right)$
(D) $\sin^{-1}\left(\frac{1}{\sqrt{6}}\right)$

Q192·MathematicsSingle correctJEE Main 2021

Let O be the origin. Let

\overrightarrow{OP} = x\hat{i}+y\hat{j}-\hat{k}

and

\overrightarrow{OQ} = -\hat{i}+2\hat{j}+3x\hat{k}

,

x, y \in R

,

x > 0

, be such that

\left|\overrightarrow{PQ}\right| = \sqrt{20}

and the vector

\overrightarrow{OP}

is perpendicular to

\overrightarrow{OQ}

. If

\overrightarrow{OR} = 3\hat{i}+z\hat{j}-7\hat{k}

,

z \in R

, is coplanar with

\overrightarrow{OP}

and

\overrightarrow{OQ}

, then the value of

x^{2}+y^{2}+z^{2}

is equal to

(A) $7$
(B) $9$
(C) $2$
(D) $1$

Q193·MathematicsSingle correctJEE Main 2021

Let

\vec{a} = 2\hat{i} - 3\hat{j} + 4\hat{k}

and

\vec{b} = 7\hat{i} + \hat{j} - 6\hat{k}

. If

\vec{r} \times \vec{a} = \vec{r} \times \vec{b}, \vec{r} \cdot \left(\hat{i} + 2\hat{j} + \hat{k}\right) = -3

, then

\vec{r} \cdot \left(2\hat{i} - 3\hat{j} + \hat{k}\right)

is equal to :

(A)12
(B)8
(C)13
(D)10

Q194·MathematicsNumericalJEE Main 2021

Let

\vec{x}

be a vector in the plane containing vectors

\vec{a} = 2\hat{i}-\hat{j}+\hat{k}

and

\vec{b} = \hat{i}+2\hat{j}-\hat{k}

. If the vector

\vec{x}

is perpendicular to

\left(3\hat{i}+2\hat{j}-\hat{k}\right)

and its projection on

\vec{a}

is

\frac{17\sqrt{6}}{2}

, then the value of

\left|\vec{x}\right|^{2}

is equal to ________.

Q195·MathematicsNumericalJEE Main 2021

If

\vec{a} = \alpha\hat{i} + \beta\hat{j} + 3\hat{k}

,

\vec{b} = -\beta\hat{i} - \alpha\hat{j} - \hat{k}

and

\vec{c} = \hat{i} - 2\hat{j} - \hat{k}

such that

\vec{a} \cdot \vec{b} = 1

and

\vec{b} \cdot \vec{c} = -3

, then

\frac{1}{3}\left(\left(\vec{a} \times \vec{b}\right) \cdot \vec{c}\right)

is equal to _______.

Q196·MathematicsSingle correctJEE Main 2021

Let

\vec{a}=\hat{i}+2\hat{j}-3\hat{k}

and

\vec{b}=2\hat{i}-3\hat{j}+5\hat{k}

. If

\vec{r}\times\vec{a}=\vec{b}\times\vec{r}

,

\vec{r}.\left(\alpha\hat{i}+2\hat{j}+\hat{k}\right)=3

and

\vec{r}.\left(2\hat{i}+5\hat{j}-\alpha\hat{k}\right)=-1

,

\alpha \in R

, then the value of

\alpha+\left|\vec{r}\right|^{2}

is equal to :

(A) $9$
(B) $15$
(C) $13$
(D) $11$

Q197·MathematicsSingle correctJEE Main 2021

Let a vector

\alpha\hat{i} + \beta\hat{j}

be obtained by rotating the vector

\sqrt{3}\hat{i} + \hat{j}

by an angle 45° about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices (α, β), (0, β) and (0, 0) is equal to

(A) $\frac{1}{2}$
(B)1
(C) $\frac{1}{\sqrt{2}}$
(D) $2\sqrt{2}$

Q198·MathematicsNumericalJEE Main 2021

Let

\vec{c}

be a vector perpendicular to the vectors

\vec{a}=\hat{i}+\hat{j}-\hat{k}

and

\vec{b}=\hat{i}+2\hat{j}+\hat{k}

. If

\vec{c}.\left(\hat{i}+\hat{j}+3\hat{k}\right)=8

then the value of

\vec{c}.\left(\vec{a}\times\vec{b}\right)

is equal to _________.

Q199·MathematicsSingle correctJEE Main 2021

Let the position vectors of two points P and Q be

3\hat{i} - \hat{j} + 2\hat{k}

and

\hat{i} + 2\hat{j} - 4\hat{k}

, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, –1, 2) and (–2, 1, –2), respectively. Let lines PR and QS intersect at T. If the vector

\overrightarrow{TA}

is perpendicular to both

\overrightarrow{PR}

and

\overrightarrow{QS}

and the length of vector

\overrightarrow{TA}

is

\sqrt{5}

units, then the modulus of a position vector of A is :

(A) $\sqrt{482}$
(B) $\sqrt{171}$
(C) $\sqrt{5}$
(D) $\sqrt{227}$

Q200·MathematicsSingle correctJEE Main 2021

If vectors

\vec{a}_{1} = x\hat{i} - \hat{j} + \hat{k}

and

\vec{a}_{2} = \hat{i} + y\hat{j} + z\hat{k}

are collinear, then a possible unit vector parallel to the vector

x\hat{i} + y\hat{j} + z\hat{k}

is :

(A) $\frac{1}{\sqrt{2}} (-\hat{j} + \hat{k})$
(B) $\frac{1}{\sqrt{2}} (\hat{i} - \hat{j})$
(C) $\frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k})$
(D) $\frac{1}{\sqrt{3}} (\hat{i} + \hat{j} - \hat{k})$

Q201·MathematicsSingle correctJEE Main 2021

If

\vec{a}

&

\vec{b}

are perpendicular vactors, then

\vec{a} \times \left( \vec{a} \times \left( \vec{a} \times \left( \vec{a} \times \vec{b} \right) \right) \right)

is equal to

(A) $\frac{1}{2}\left|\vec{a}\right|^{4}\,\vec{b}$
(B) $\vec{a} \times \vec{b}$
(C) $\left|\vec{a}\right|^{4}\,\vec{b}$
(D) $\vec{0}$

Q202·MathematicsNumericalJEE Main 2021

Let a=ˆi+α+ˆj3kˆ andb=3iˆ−α+ˆjkˆ . If the area of the parallelogram whose adjacent sides are represented by the vectors a and b is 8 3 square units, then a·b is equal to ______.

Q203·MathematicsNumericalJEE Main 2021

Let

\vec{a} = \hat{i} + 2\hat{j} - \hat{k}

,

\vec{b} = \hat{i} - \hat{j}

and

\vec{c} = \hat{i} - \hat{j} - \hat{k}

be three given vectors. If

\vec{r}

is a vector such that

\vec{r} \times \vec{a} = \vec{c} \times \vec{a}

and

\vec{r} \cdot \vec{b} = 0

, then

\vec{r} \cdot \vec{a}

is equal to ________

Q204·MathematicsNumericalJEE Main 2021

Let three vectors

\vec{a}, \vec{b}

and

\vec{c}

be such that

\vec{c}

is coplanar with

\vec{a}

and

\vec{b}

,

\vec{a} \cdot \vec{c} = 7

and

\vec{b}

is perpendicular to

\vec{c}

, where

\vec{a} = -\hat{i} + \hat{j} + \hat{k}

and

\vec{b} = 2\hat{i} + \hat{k}

, then the value of

2\left|\vec{a} + \vec{b} + \vec{c}\right|^2

is ________

Q205·MathematicsMultiple correctJEE Advanced 2020

Let

a

and

b

be positive real numbers. Suppose

\overrightarrow{PQ} = a\hat{i} + b\hat{j}

and

\overrightarrow{PS} = a\hat{i} - b\hat{j}

are adjacent sides of a parallelogram

PQRS

. Let

\vec{u}

and

\vec{v}

be the projection vectors of

\vec{w} = \hat{i} + \hat{j}

along

\overrightarrow{PQ}

and

\overrightarrow{PS}

, respectively. If

|\vec{u}| + |\vec{v}| = |\vec{w}|

and if the area of the parallelogram

PQRS

is 8, then which of the following statements is/are TRUE?

(A) $a + b = 4$
(B) $a - b = 2$
(C)The length of the diagonal $PR$ of the parallelogram $PQRS$ is 4
(D) $\vec{w}$ is an angle bisector of the vectors $\overrightarrow{PQ}$ and $\overrightarrow{PS}$

Q206·MathematicsNumericalJEE Advanced 2020

In a triangle PQR, let

\vec{a} = \overrightarrow{QR}

,

\vec{b} = \overrightarrow{RP}

and

\vec{c} = \overrightarrow{PQ}

. If

|\vec{a}| = 3

,

|\vec{b}| = 4

and

\frac{\vec{a}\cdot(\vec{c} - \vec{b})}{\vec{c}\cdot(\vec{a} - \vec{b})} = \frac{|\vec{a}|}{|\vec{a}| + |\vec{b}|}

, then the value of

\left|\vec{a} \times \vec{b}\right|^{2}

is ______

Q207·MathematicsNumericalJEE Main 2020

If

\vec{a}

and

\vec{b}

are unit vectors, then the greatest value of

\sqrt{3}\left|\vec{a}+\vec{b}\right|+\left|\vec{a}-\vec{b}\right|

is ____.

Q208·MathematicsNumericalJEE Main 2020

If

\vec{x}

and

\vec{y}

be two non-zero vectors such that

|\vec{x} + \vec{y}| = |\vec{x}|

and

2\vec{x} + \lambda\vec{y}

is perpendicular to

\vec{y}

, then the value of

\lambda

is

Q209·MathematicsSingle correctJEE Main 2020

If the volume of a parallelepiped, whose coterminus edges are given by the vectors

\vec{a}=\hat{i}+\hat{j}+n\hat{k}, \vec{b}=2\hat{i}+4\hat{j}-n\hat{k}

and

\vec{c}=\hat{i}+n\hat{j}+3\hat{k}

(n \ge 0)

, is 158 cu.units. then:

(A) $\vec{a}.\vec{c}=14$
(B)n=7
(C) $\vec{b}.\vec{c}=10$
(D)n=9

Q210·MathematicsNumericalJEE Main 2020

Let the vectors

\vec{a}, \vec{b}, \vec{c}

be such that

|\vec{a}| = 2, |\vec{b}| = 4

and

|\vec{c}| = 4

. if the projection of

\vec{b}

on

\vec{a}

is equal to the projection of

\vec{c}

on

\vec{a}

and

\vec{b}

is perpendicular to

\vec{c}

, then the value of

|\vec{a} + \vec{b} - \vec{c}|

is __________.

Q211·MathematicsSingle correctJEE Main 2020

Let

x_{0}

be the point of local maxima of

f(x) = \vec{a} \cdot \left( \vec{b} \times \vec{c} \right)

where

\vec{a} = x\hat{i} - 2\hat{j} + 3\hat{k}

,

\vec{b} = -2\hat{i} + x\hat{j} - \hat{k}

and

\vec{c} = 7\hat{i} - 2\hat{j} + x\hat{k}

. Then the value of

\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}

at

x = x_{0}

is

(A) $-4$
(B) $-22$
(C) $-30$
(D) $14$

Q212·MathematicsNumericalJEE Main 2020

If

\vec{a} = 2\hat{i} + \hat{j} + 2\hat{k}

, then the value of

\left|\hat{i} \times \left(\vec{a} \times \hat{i}\right)\right|^{2} + \left|\hat{j} \times \left(\vec{a} \times \hat{j}\right)\right|^{2} + \left|\hat{k} \times \left(\vec{a} \times \hat{k}\right)\right|^{2}

is equal to:

Q213·MathematicsSingle correctJEE Main 2020

Let

a, b, c \in R

be such that

a^2 + b^2 + c^2 = 1

. If

a\cos\theta = b\cos\left(\theta + \frac{2\pi}{3}\right) = c\cos\left(\theta + \frac{4\pi}{3}\right)

, where

\theta = \frac{\pi}{9}

, then the angle between the vectors

a\hat{i} + b\hat{j} + c\hat{k}

and

b\hat{i} + c\hat{j} + a\hat{k}

is:

(A) $\frac{\pi}{9}$
(B) $\frac{2\pi}{3}$
(C) $0$
(D) $\frac{\pi}{2}$

Q214·MathematicsNumericalJEE Main 2020

If the vectors,

\vec{p}=(a+1)\hat{i}+a\hat{j}+a\hat{k}

,

\vec{q}=a\hat{i}+(a+1)\hat{j}+a\hat{k}

and

\vec{r}=a\hat{i}+a\hat{j}+(a+1)\hat{k}

(a \in R)

are coplanar and

3(\vec{p}.\vec{q})^{2}-\lambda|\vec{r}\times\vec{q}|^{2}=0

, then the value of

\lambda

is ______.

Q215·MathematicsNumericalJEE Main 2020

Let

\vec{a}, \vec{b}

and

\vec{c}

be three vectors such that

|\vec{a}| = \sqrt{3}, |\vec{b}| = 5, \vec{b}.\vec{c} = 10

and the angle between

\vec{b}

and

\vec{c}

is

\frac{\pi}{3}

. If

\vec{a}

is perpendicular to the vector

\vec{b} \times \vec{c}

, then

|\vec{a} \times (\vec{b} \times \vec{c})|

is equal to ____.

Q216·MathematicsSingle correctJEE Main 2020

Let

\vec{a}=\hat{i}-2\hat{j}+\hat{k}

and

\vec{b}=\hat{i}-\hat{j}+\hat{k}

be two vectors. If

\vec{c}

is a vector such that

\vec{b}\times\vec{c}=\vec{b}\times\vec{a}

and

\vec{c}.\vec{a}=0

, then

\vec{c}.\vec{b}

is equal to:

(A) $-\dfrac{1}{2}$
(B) $-\dfrac{3}{2}$
(C) $-1$
(D) $\dfrac{1}{2}$

Q217·MathematicsSingle correctJEE Main 2020

Let the volume of a parallelepiped whose coterminous edges are given by

\vec{u}=\hat{i}+\hat{j}+\lambda\hat{k}

,

\vec{v}=\hat{i}+\hat{j}+3\hat{k}

and

\vec{w}=2\hat{i}+\hat{j}+\hat{k}

be 1 cu. Unit. If

\theta

be the angle between the edges

\vec{u}

and

\vec{w}

, then

\cos\theta

can be:

(A) $\dfrac{7}{6\sqrt{3}}$
(B) $\dfrac{5}{7}$
(C) $\dfrac{5}{3\sqrt{3}}$
(D) $\dfrac{7}{6\sqrt{6}}$

Q218·MathematicsSingle correctJEE Main 2020

A vector

\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k} \ (\alpha,\beta \in R)

lies in the plane of the vectors,

\vec{b} = \hat{i} + \hat{j}

and

\vec{c} = \hat{i} - \hat{j} + 4\hat{k}

. If

\vec{a}

bisects the angle between

\vec{b}

and

\vec{c}

, then:

(A) $\vec{a}.\hat{k} + 4 = 0$
(B) $\vec{a}.\hat{k} + 2 = 0$
(C) $\vec{a}.\hat{i} + 3 = 0$
(D) $\vec{a}.\hat{i} + 1 = 0$

Q219·MathematicsNumericalJEE Advanced 2019

Let

\vec{a} = 2\hat{i} + \hat{j} - \hat{k}

and

\vec{b} = \hat{i} + 2\hat{j} + \hat{k}

be two vectors. Consider a vector

\vec{c} = \alpha\vec{a} + \beta\vec{b}

,

\alpha,\ \beta \in \mathbb{R}

. If the projection of

\vec{c}

on the vector

\left(\vec{a} + \vec{b}\right)

is

3\sqrt{2}

, then the minimum value of

\left(\vec{c} - \left(\vec{a} \times \vec{b}\right)\right)\cdot\vec{c}

equals ____

Q220·MathematicsSingle correctJEE Main 2019

Let

\vec{a} = 3\hat{i} + 2\hat{j} + 2\hat{k}

and

\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}

be two vectors. If a vector perpendicular to both the vectors

\vec{a} + \vec{b}

and

\vec{a} - \vec{b}

has the magnitude 12 then one such vector is:

(A) $4\left(2\hat{i} - 2\hat{j} - \hat{k}\right)$
(B) $4\left(2\hat{i} - 2\hat{j} + \hat{k}\right)$
(C) $4\left(2\hat{i} + 2\hat{j} + \hat{k}\right)$
(D) $4\left(2\hat{i} + 2\hat{j} - \hat{k}\right)$

Q221·MathematicsSingle correctJEE Main 2019

Let

\alpha \in R

and the three vectors

\vec{a} = \alpha\hat{i} + \hat{j} + 3\hat{k}

,

\vec{b} = 2\hat{i} + \hat{j} - \alpha\hat{k}

and

\vec{c} = \alpha\hat{i} - 2\hat{j} + 3\hat{k}

. Then the set

S = (\alpha : \vec{a}, \vec{b} \text{ and } \vec{c} \text{ are coplanar})

(A)Contains exactly two numbers only one of which is positive
(B)is empty
(C)Contains exactly two positive numbers
(D)is singleton

Q222·MathematicsSingle correctJEE Main 2019

Let A(3, 0, −1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then cos(

\angle

GOA) (O being he origin) is equal to:

(A) $\dfrac{1}{\sqrt{30}}$
(B) $\dfrac{1}{2\sqrt{15}}$
(C) $\dfrac{1}{6\sqrt{10}}$
(D) $\dfrac{1}{\sqrt{15}}$

Q223·MathematicsSingle correctJEE Main 2019

If a unit vector

\vec{r}

makes angles

\dfrac{\pi}{3}

with

\hat{i}

,

\dfrac{\pi}{4}

with

\hat{j}

and

\theta\in\left(0,\pi\right)

with

\hat{k}

, then a value of

\theta

is

(A) $\dfrac{5\pi}{12}$
(B) $\dfrac{5\pi}{6}$
(C) $\dfrac{2\pi}{3}$
(D) $\dfrac{\pi}{4}$

Q224·MathematicsSingle correctJEE Main 2019

Let

\vec{\alpha}=3\hat{i}+\hat{j}

and

\vec{\beta}=2\hat{i}-\hat{j}+3\hat{k}

. If

\vec{\beta}=\vec{\beta_{1}}-\vec{\beta_{2}}

, where

\vec{\beta_{1}}

is parallel to

\vec{\alpha}

and

\vec{\beta_{2}}

is perpendicular to

\vec{\alpha}

, then

\vec{\beta_{1}}\times\vec{\beta_{2}}

is equal to:

(A) $\dfrac{1}{2}(-3\hat{i}+9\hat{j}+5\hat{k})$
(B) $\dfrac{1}{2}(3\hat{i}-9\hat{j}+5\hat{k})$
(C) $-3\hat{i}+9\hat{j}+5\hat{k}$
(D) $3\hat{i}-9\hat{j}-5\hat{k}$

Q225·MathematicsSingle correctJEE Main 2019

Let

\vec{a}=3\hat{i}+2\hat{j}+x\hat{k}

and

\vec{b}=\hat{i}--\hat{j}+\hat{k}

, for some real x. Then

|\vec{a}\times\vec{b}|=r

is possible if:

(A) $r\geq5\sqrt{\frac{3}{2}}$
(B) $3\sqrt{\frac{3}{2}}<r<5\sqrt{\frac{3}{2}}$
(C) $\sqrt{\frac{3}{2}}<r\leq3\sqrt{\frac{3}{2}}$
(D) $0<r\leq\sqrt{\frac{3}{2}}$

Q226·MathematicsSingle correctJEE Main 2019

The magnitude of the projection of the vector

2\hat{i}+3\hat{j}+\hat{k}

on the vector perpendicular to the plane containing the vectors

\hat{i}+\hat{j}+\hat{k}

and

\hat{i}+2\hat{j}+3\hat{k}

, is:

(A) $3\sqrt{6}$
(B) $\frac{\sqrt{3}}{2}$
(C) $\sqrt{6}$
(D) $\sqrt{\frac{3}{2}}$

Q227·MathematicsSingle correctJEE Main 2019

The sum of the distinct real values of

\mu

, for which the vectors,

\mu\hat{i} + \hat{j} + \hat{k}

,

\hat{i} + \mu\hat{j} + \hat{k}

,

\hat{i} + \hat{j} + \mu\hat{k}

are co-planar, is:

(A)-1
(B)0
(C)1
(D)2

Q228·MathematicsSingle correctJEE Main 2019

Let

\vec{a}

,

\vec{b}

, and

\vec{c}

be three unit vectors, out of which vectors

\vec{b}

and

\vec{c}

are non-parallel. If α and β are the angles which vector

\vec{a}

makes with vectors

\vec{b}

and

\vec{c}

respectively and

\vec{a} \times (\vec{b} \times \vec{c}) = \dfrac{1}{2}\vec{b}

, then |α − β| is equal to :

(A) $30°$
(B) $90°$
(C) $60°$
(D) $45°$

Q229·MathematicsSingle correctJEE Main 2019

Let

\vec{a}=\hat{i}+2\hat{j}+4\hat{k}, \vec{b}=\hat{i}+\lambda\hat{j}+4\hat{k}

and

\vec{c}=2\hat{i}+4\hat{j}+\left(\lambda^{2}-1\right)\hat{k}

be coplanar vectors. Then the non − zero vector

\vec{a}\times\vec{c}

is:

(A) $-10\hat{i}-5\hat{j}$
(B) $-14\hat{i}-5\hat{j}$
(C) $-14\hat{i}+5\hat{j}$
(D) $-10\hat{i}+5\hat{j}$

Q230·MathematicsSingle correctJEE Main 2019

Let

\vec{a} = 2\hat{i} + \lambda_1\hat{j} + 3\hat{k}

,

\vec{b} = 4\hat{i} + (3 - \lambda_2)\hat{j} + 6\hat{k}

and

\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_3 - 1)\hat{k}

be three vectors such that

\vec{b} = 2\vec{a}

and

\vec{a}

is perpendicular to

\vec{c}

. Then a possible value of

(\lambda_1, \lambda_2, \lambda_3)

is:

(A)(1, 3, 1)
(B) $\left(-\dfrac{1}{2}, 4, 0\right)$
(C) $\left(\dfrac{1}{2}, 4, -2\right)$
(D)(1, 5, 1)

Q231·MathematicsSingle correctJEE Main 2019

Let

\vec{a} = \hat{i} + \hat{j} + \sqrt{2}\hat{k}, \vec{b} = b_{1}\hat{i} + b_{2}\hat{j} + \sqrt{2}\hat{k}

and

\vec{c} = 5\hat{i} + \hat{j} + \sqrt{2}\hat{k}

be three vectors such that the projection vector of

\vec{b}

on

\vec{a}

is

\vec{a}

. If

\vec{a} + \vec{b}

is perpendicular to

\vec{c}

, then

\left| \vec{b} \right|

is equal to:

(A) $\sqrt{22}$
(B) $4$
(C) $\sqrt{32}$
(D) $6$

Q232·MathematicsSingle correctJEE Main 2019

Let

\vec a = \hat i - \hat j

,

\vec b = \hat i + \hat j + \hat k

and

\vec c

be a vector such that

\vec a \times \vec c + \vec b = 0

and

\vec a . \vec c = 4

, then

|\vec c|^2

is equal to:

(A) $\dfrac{19}{2}$
(B)9
(C)8
(D) $\dfrac{17}{2}$

Q233·MathematicsNumericalJEE Advanced 2018

Consider the cube in the first octant with sides OP, OQ and OR of length 1, along the x-axis, y-axis and z-axis, respectively, where O(0, 0, 0) is the origin. Let

S\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)

be the centre of the cube and T be the vertex of the cube opposite to the origin O such that S lies on the diagonal OT. If

\vec{p} = \overrightarrow{SP}

,

\vec{q} = \overrightarrow{SQ}

,

\vec{r} = \overrightarrow{SR}

and

\vec{t} = \overrightarrow{ST}

, then the value of

\left|(\vec{p} \times \vec{q}) \times (\vec{r} \times \vec{t})\right|

is ______ .

Q234·MathematicsNumericalJEE Advanced 2018

Let

\vec{a}

and

\vec{b}

be two unit vectors such that

\vec{a} \cdot \vec{b} = 0

. For some x, y

\in

R, let

\vec{c} = x\vec{a} + y\vec{b} + \left(\vec{a} \times \vec{b}\right)

. If

|\vec{c}| = 2

and the vector

\vec{c}

is inclined at the same angle

\alpha

to both

\vec{a}

and

\vec{b}

, then the value of

8\cos^{2}\alpha

is ______ .

Q235·MathematicsSingle correctJEE Advanced 2017

Let

O

be the origin and let

PQR

be an arbitrary triangle. The point

S

is such that

\overrightarrow{OP}\cdot\overrightarrow{OQ} + \overrightarrow{OR}\cdot\overrightarrow{OS} = \overrightarrow{OR}\cdot\overrightarrow{OP} + \overrightarrow{OQ}\cdot\overrightarrow{OS} = \overrightarrow{OQ}\cdot\overrightarrow{OR} + \overrightarrow{OP}\cdot\overrightarrow{OS}

Then the triangle

PQR

has

S

as its

(A)centroid
(B)circumcentre
(C)incentre
(D)orthocenter

Q236·MathematicsSingle correctJEE Advanced 2017

PARAGRAPH 1 Let

O

be the origin, and

\overrightarrow{OX}

,

\overrightarrow{OY}

,

\overrightarrow{OZ}

be three unit vectors in the directions of the sides

\overrightarrow{QR}

,

\overrightarrow{RP}

,

\overrightarrow{PQ}

, respectively, of a triangle

PQR

.

\left|\overrightarrow{OX} \times \overrightarrow{OY}\right| =

(A) $\sin(P + Q)$
(B) $\sin 2R$
(C) $\sin(P + R)$
(D) $\sin(Q + R)$

Q237·MathematicsMultiple correctJEE Advanced 2016

Let

\hat{u} = u_{1}\hat{i} + u_{2}\hat{j} + u_{3}\hat{k}

be a unit vector in

\mathbb{R}^{3}

and

\hat{w} = \frac{1}{\sqrt{6}} \left( \hat{i} + \hat{j} + 2\hat{k} \right)

. Given that there exists a vector

\vec{v}

in

\mathbb{R}^{3}

such that

|\hat{u} \times \vec{v}| = 1

and

\hat{w} \cdot (\hat{u} \times \vec{v}) = 1

. Which of the following statement(s) is(are) correct?

(A)There is exactly one choice for such $\vec{v}$
(B)There are infinitely many choices for such $\vec{v}$
(C)If $\hat{u}$ lies in the xy-plane then $|u_{1}| = |u_{2}|$
(D)If $\hat{u}$ lies in the xz-plane then $2|u_{1}| = |u_{3}|$

Q238·MathematicsMultiple correctJEE Advanced 2015

Let

\Delta PQR

be a triangle. Let

\vec{a} = \overrightarrow{QR}

,

\vec{b} = \overrightarrow{RP}

and

\vec{c} = \overrightarrow{PQ}

. If

|\vec{a}| = 12

,

|\vec{b}| = 4\sqrt{3}

and

\vec{b} \cdot \vec{c} = 24

, then which of the following is (are) true ?

(A) $\dfrac{|\vec{c}|^{2}}{2} - |\vec{a}| = 12$
(B) $\dfrac{|\vec{c}|^{2}}{2} + |\vec{a}| = 30$
(C) $|\vec{a} \times \vec{b} + \vec{c} \times \vec{a}| = 48\sqrt{3}$
(D) $\vec{a} \cdot \vec{b} = -72$

Q239·MathematicsIntegerJEE Advanced 2015

Suppose that

\vec{p}

,

\vec{q}

and

\vec{r}

are three non-coplanar vectors in

\mathbb{R}^{3}

. Let the components of a vector

\vec{s}

along

\vec{p}

,

\vec{q}

and

\vec{r}

be 4, 3 and 5, respectively. If the components of this vector

\vec{s}

along

\left(-\vec{p} + \vec{q} + \vec{r}\right)

,

\left(\vec{p} - \vec{q} + \vec{r}\right)

and

\left(-\vec{p} - \vec{q} + \vec{r}\right)

are

x

,

y

and

z

, respectively, then the value of

2x + y + z

is

Q240·MathematicsMatrix matchJEE Advanced 2015

Match the entries in Column I with the entries in Column II.

Column – I	Column – II
A.In $\mathbb{R}^{2}$ , if the magnitude of the projection vector of the vector $\alpha\hat{i} + \beta\hat{j}$ on $\sqrt{3}\hat{i} + \hat{j}$ is $\sqrt{3}$ and if $\alpha = 2 + \sqrt{3}\beta$ , then possible value(s) of $\|\alpha\|$ is (are)	P.1
B.Let $a$ and $b$ be real numbers such that the function $f(x) = \begin{cases} -3ax^{2} - 2, & x < 1 \\ bx + a^{2}, & x \ge 1 \end{cases}$ is differentiable for all $x \in \mathbb{R}$ . Then possible value(s) of $a$ is (are)	Q.2
C.Let $\omega \ne 1$ be a complex cube root of unity. If $(3 - 3\omega + 2\omega^{2})^{4n+3} + (2 + 3\omega - 3\omega^{2})^{4n+3} + (-3 + 2\omega + 3\omega^{2})^{4n+3} = 0$ , then possible value(s) of $n$ is (are)	R.3
D.Let the harmonic mean of two positive real numbers $a$ and $b$ be 4. If $q$ is a positive real number such that $a$ , 5, $q$ , $b$ is an arithmetic progression, then the value(s) of $\|q - a\|$ is (are)	S.4
	T.5

Q241·MathematicsMultiple correctJEE Advanced 2014

Let

\vec{x}

,

\vec{y}

and

\vec{z}

be three vectors each of magnitude

\sqrt{2}

and the angle between each pair of them is

\frac{\pi}{3}

. If

\vec{a}

is a non-zero vector perpendicular to

\vec{x}

and

\vec{y} \times \vec{z}

and

\vec{b}

is a non-zero vector perpendicular to

\vec{y}

and

\vec{z} \times \vec{x}

, then

(A) $\vec{b} = (\vec{b} \cdot \vec{z})(\vec{z} - \vec{x})$
(B) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{y} - \vec{z})$
(C) $\vec{a} \cdot \vec{b} = -(\vec{a} \cdot \vec{y})(\vec{b} \cdot \vec{z})$
(D) $\vec{a} = (\vec{a} \cdot \vec{y})(\vec{z} - \vec{y})$

Q242·MathematicsIntegerJEE Advanced 2014

Let

\vec{a}

,

\vec{b}

, and

\vec{c}

be three non-coplanar unit vectors such that the angle between every pair of them is

\frac{\pi}{3}

. If

\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p\vec{a} + q\vec{b} + r\vec{c}

, where

p

,

q

and

r

are scalars, then the value of

\frac{p^2 + 2q^2 + r^2}{q^2}

is __________

Q243·MathematicsIntegerJEE Advanced 2013

Consider the set of eight vectors

V=\{a\hat{i}+b\hat{j}+c\hat{k}; a,b,c\in\{-1,1\}\}

. Three non-coplanar vectors can be chosen from V in

2^p

ways. Then p is ________

Q244·MathematicsSingle correctJEE Advanced 2013

Match List I with List II and select the correct answer using the code given below the lists :

List-I	List-II
P.Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is 2. Then the volume of the parallelepiped determined by vectors $2\left(\vec{a} \times \vec{b}\right), 3\left(\vec{b} \times \vec{c}\right)$ and $\left(\vec{c} \times \vec{a}\right)$ is	1. $100$
Q.Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is 5. Then the volume of the parallelepiped determined by vectors $3\left(\vec{a} + \vec{b}\right), \left(\vec{b} + \vec{c}\right)$ and $2\left(\vec{c} + \vec{a}\right)$ is	2. $30$
R.Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is 20. Then the area of the triangle with adjacent sides determined by vectors $\left(2\vec{a} + 3\vec{b}\right)$ and $\left(\vec{a} - \vec{b}\right)$ is	3. $24$
S.Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is 30. Then the area of the parallelogram with adjacent sides determined by vectors $\left(\vec{a} + \vec{b}\right)$ and $\vec{a}$ is	4. $60$

(A)P-4, Q-2, R-3, S-1
(B)P-2, Q-3, R-1, S-4
(C)P-3, Q-4, R-1, S-2
(D)P-1, Q-4, R-3, S-2

Q245·MathematicsSingle correctJEE Advanced 2013

Let

\overrightarrow{PR}=3\hat{i}+\hat{j}-2\hat{k}

and

\overrightarrow{SQ}=\hat{i}-3\hat{j}-4\hat{k}

determine diagonals of a parallelogram PQRS and

\overrightarrow{PT}=\hat{i}+2\hat{j}+3\hat{k}

be another vector. Then the volume of the parallelepiped determined by the vectors

\overrightarrow{PT},\overrightarrow{PQ}

and

\overrightarrow{PS}

is

(A) $5$
(B) $20$
(C) $10$
(D) $30$

Vector Algebra — JEE Previous Year Questions

All 245 Vector Algebra questions

Vector Algebra — frequently asked

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Practise Vector Algebra until it stops costing you marks.