# Vector Algebra JEE Main Previous year Questions with solution

On this page you will get Vector Algebra JEE Main Previous year Questions with complete and detailed solution. These solutions are arranged in chronological order.

## Vector Algebra JEE Main Previous year Questions with solution

2020

Q. Let $a, b, c \in \mathbf{R}$ 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) $\frac{\pi}{2}$

(b) $\frac{2 \pi}{3}$

(c) $\frac{\pi}{9}$

(d) 0

[Sep. 03, 2020 (II)]

Sol.

Q. Let the position vectors of points ‘A’ and ‘B’ be $\hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}+\hat{j}+3 \hat{k}$, respectively. A point ‘P’ divides the line segment $\mathrm{AB}$ internally in the ratio $\lambda: 1(\lambda>0)$. If $O$ is the region and $\overrightarrow{O B} \cdot \overrightarrow{O P}-3|\overrightarrow{O A} \times \overrightarrow{O P}|^{2}=6$, then $\lambda$ is equal to

[NA Sep. 02,2020 (II)]

Sol.

Q. 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 $\vec{r}=a \hat{i}+a \hat{j}+(a+1) \hat{k} \quad(a \in \mathrm{R})$ are coplanar and $3(\vec{p} \cdot \vec{q})^{2}-\lambda|\vec{r} \times \vec{q}|^{2}=0$, then the value of $\lambda$ is

[NA Jan. 9, 2020 (I)]

Sol.

Q. If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is

[NA Sep. 06, 2020 (I)]

Sol.

Q. If $\rightarrow \underset{\rightarrow}{\rightarrow}$ and $y$ 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

[NA Sep. 06, 2020 (II)]

Sol.

Q. 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

[NA Sep. 05, 2020 (II)]

Sol.

Q. Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^{2}+|\vec{a}-\vec{c}|^{2}=8$. Then $|\vec{a}+2 \vec{b}|^{2}+|\vec{a}+2 \vec{c}|^{2}$ is equal to

[NA Sep. 02, 2020 (I)]

Sol.

Q. The projection of the line segment joining the points $(1,-1,3)$ and $(2,-4,11)$ on the line joining the points $(-1,2,3)$ and $(3,-2,10)$ is

[NA Jan. 9,2020(I)]

Sol.

Q. Let the volume of a parallelopiped 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) $\frac{7}{6 \sqrt{6}}$

(b) $\frac{7}{6 \sqrt{3}}$

(c) $\frac{5}{7}$

(d) $\frac{5}{3 \sqrt{3}}$

[Jan. 8, 2020 (I)]

Sol.

Q. A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in \boldsymbol{R})$ 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) $\vec{a} \cdot \hat{i}+3=0$

(b) $\vec{a} \cdot \hat{i}+1=0$

(c) $\vec{a} \cdot \hat{k}+2=0$

(d) $\vec{a} \cdot \hat{k}+4=0$

[Jan. 7, 2020 (I)]

Sol.

Q. If the volume of a parallelopiped, 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 $\vec{c}=\hat{i}+n \hat{j}+3 \hat{k}(n \geq 0)$, is 158 cu.units, then:

[Sep. 05, 2020 (I)]

Sol.

Q. Let $x_{0}$ be the point of local maxima of $f(x)=a \cdot(b \times c)$, where $\vec{a}=x \hat{i}-2 \hat{j}+3 \hat{k}, \quad \vec{b}=-2 \hat{i}+x \hat{j}-\hat{k}$ 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) $-30$

(c) 14

(d) $-22$

[Sep. 04, $2020(\mathrm{I})]$

Sol.

Q. If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of

$|\hat{i} \times(\vec{a} \times \hat{i})|^{2}+|\hat{j} \times(\vec{a} \times \hat{j})|^{2}+|\hat{k} \times(\vec{a} \times \hat{k})|^{2}$ is equal to

[NA Sep. 04, 2020 (II)]

Sol.

Q. Let $\vec{b}, \vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3},|\vec{b}|=5, \vec{b} \cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{3}$. If $\vec{b}$ is perpendicular to the vector $\vec{b} \times \vec{c}$, then $|\vec{a} \times(\vec{b} \times \vec{c})|$ is equal to

[NA Jan. 9, 2020 (II)]

Sol.

Q. 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 $\vec{c} \cdot \vec{a}=0$, then $\vec{c} \cdot \vec{b}$ is equal to:

(a) $-\frac{3}{2}$

(b) $\frac{1}{2}$

(c) $-\frac{1}{2}$

(d) $-1$

[Jan. 8, 2020 (II)]

Sol.

Q. Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$. if $\lambda=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ and $\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}$, then the ordered pair, $(\lambda, \vec{d})$ is equal to:

(a) $\left(\frac{3}{2}, 3 \vec{a} \times \vec{c}\right)$

(b) $\left(-\frac{3}{2}, 3 \vec{c} \times \vec{b}\right)$

(c) $\left(\frac{3}{2}, 3 \vec{b} \times \vec{c}\right)$

(d) $\left(-\frac{3}{2}, 3 \vec{a} \times \vec{b}\right)$

[Jan. 7, 2020 (II)]

Sol.

### 2019

Q. 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) $4(2 \hat{i}+2 \hat{j}+2 \hat{k})$

(b) $4(2 \hat{i}-2 \hat{j}-\hat{k})$

(c) $4(2 \hat{i}+2 \hat{j}-\hat{k})$

(d) $4(-2 \hat{i}-2 \hat{j}+\hat{k})$

[April 12, 2019 (I)]

Sol.

Q. If the volume of parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}, \hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

(a) $-\frac{1}{\sqrt{3}}$

(b) $\frac{1}{\sqrt{3}}$

(c) $\sqrt{3}$

(d) $-\sqrt{3}$

[April 12, 2019 (I)]

Sol.

Q. Let $\alpha \in \mathrm{R}$ and the three vectors $\vec{a}=\alpha \hat{i}+\hat{j}+3 \hat{k}$, $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k} \quad$ and $\vec{c}=\alpha \hat{i}-2 \hat{j}+3 \hat{k} .$ Then the set $S=(\alpha: \vec{a}, \vec{b}$ and $\vec{c}$ are coplanar) $\quad$

(a) is singleton

(b) is empty

(c) contains exactly two positive numbers

(d) contains exactly two numbers only one of which is positive

[April 12, 2019(II)]

Sol.

Q. If a unit vector $\vec{a}$ makes angles $\pi / 3$ with $\hat{i}, \pi / 4$ with $\hat{j}$ and $\theta \in(0, \pi)$ with $\hat{k}$, then a value of, is:

(a) $\frac{5 \pi}{6}$

(b) $\frac{\pi}{4}$

(c) $\frac{5 \pi}{12}$

(d) $\frac{2 \pi}{3}$

[April 09, 2019(II)]

Sol.

Q. 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

[Jan. 12, 2019 (I)]

Sol.

Q. 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) $-10 \hat{i}+5 \hat{j}$

[Jan. 11, 2019(I)]

Sol.

Q. Let $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\beta \hat{i}+(1-\beta) \hat{j}$ respectively be the position vectors of the points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ with respect to the origin $\mathrm{O}$. If the distance of $\mathrm{C}$ from the bisector of the acute angle between $\mathrm{OA}$ and $\mathrm{OB}$ is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :

(a) 4

(b) 3

(c) 2

(d) 1

[Jan. 11, 2019 (II)]

Sol.

Q. Let $\vec{\alpha}=(\lambda-2) \vec{a}+\vec{b}$ and $\vec{\beta}=(4 \lambda-2) \vec{a}+3 \vec{b}$ be two given vectors where vectors $\vec{a}$ and $\vec{b}$ are non-collinear. The value of $\lambda$ for which vectors $\vec{\alpha}$ and $\vec{\beta}$ are collinear, is:

(a) $-4$

(b) $-3$

(c) 4

(d) 3

[Jan. 10, 2019 (II)]

Sol.

Q. Let $\overrightarrow{\mathrm{a}}=2 \hat{i}+\lambda_{1} \hat{j}+3 \hat{k}, \overrightarrow{\mathrm{b}}=4 \hat{i}+\left(3-\lambda_{2}\right) \hat{j}+6 \hat{k}$ and $\overrightarrow{\mathrm{c}}=3 \hat{i}+6 \hat{j}+\left(\lambda_{3}-1\right) \hat{k}$ be three vectors such that $\overrightarrow{\mathrm{b}}=2 \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{a}}$ is perpendicular to $\overrightarrow{\mathrm{c}}$ Then a possible value of $\left(\lambda_{1}, \lambda_{2}, \lambda_{3}\right)$ is:

(a) $(1,3,1)$

(b) $\left(-\frac{1}{2}, 4,0\right)$

(c) $\left(\frac{1}{2}, 4,-2\right)$

(d) $(1,5,1)$

[Jan. 10, 2019 (I)]

Sol.

Q. Let $\overrightarrow{\mathrm{a}}=\hat{i}+\hat{j}+\sqrt{2} \hat{k}, \overrightarrow{\mathrm{b}}=\mathrm{b}_{1} \hat{i}+\mathrm{b}_{2} \hat{j}+\sqrt{2} \hat{k}$ and $\overrightarrow{\mathrm{c}}=5 \hat{i}+\hat{j}+\sqrt{2} \hat{k}$ 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 $|\vec{b}|$ is equal to:

(a) $\sqrt{32}$

(b) 6

(c) $\sqrt{22}$

(d) 4

[Jan. 09, 2019 (II)]

Sol.

Q. 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) $-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$

(b) $3 \hat{i}-9 \hat{j}-5 \hat{k}$

(c) $\frac{1}{2}(-3 \hat{\mathrm{i}}+9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$

(d) $\frac{1}{2}(3 \hat{\mathrm{i}}-9 \hat{\mathrm{j}}+5 \hat{\mathrm{k}})$

[April 09,2019(I)]

Sol.

Q. 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) $\frac{\sqrt{3}}{2}$

(b) $\sqrt{6}$

(c) $3 \sqrt{6}$

(d) $\sqrt{\frac{3}{2}}$

[April 08, 2019(I)]

Sol.

Q. 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 $|\vec{a} \times \vec{b}|=\mathrm{r}$ is possible if:

(a) $\sqrt{\frac{3}{2}}<r \leq 3 \sqrt{\frac{3}{2}}$

(b) $r \geq 5 \sqrt{\frac{3}{2}}$

(c) $0<r \leq \sqrt{\frac{3}{2}}$

(d) $3 \sqrt{\frac{3}{2}}<r<5 \sqrt{\frac{3}{2}}$

[April 08, 2019 (II)]

Sol.

Q. 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 $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a} \times(\vec{b} \times \vec{c})=\frac{1}{2} \vec{b}$, then $|\alpha-\beta|$ is equal to :

(a) $30^{\circ}$

(b) $90^{\circ}$

(c) $60^{\circ}$

(d) $45^{\circ}$

[Jan. 12, 2019 (II)]

Sol.

Q. Let $\vec{a}=\hat{i}-\hat{j}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c}+\vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{c}=4$, then $|\vec{c}|^{2}$ is equal to:

(a) $\frac{19}{2}$

(b) 9

(c) 8

(d) $\frac{17}{2}$

[Jan 09, 2019]

Sol.

### 2018

Q. If the position vectors of the vertices $A, B$ and $C$ of a $\triangle \mathrm{ABC}$ are respectively $4 \hat{i}+7 \hat{j}+8 \hat{k}, 2 \hat{i}+3 \hat{j}+4 \hat{k}$ and $2 \hat{i}+5 \hat{j}+7 \hat{k}$, then the position vector of the point, where the bisector of $\angle A$ meets $B C$ is

(a) $\frac{1}{2}(4 \hat{i}+8 \hat{j}+11 \hat{k})$

(b) $\frac{1}{3}(6 \hat{i}+13 \hat{j}+18 \hat{k})$

(c) $\frac{1}{4}(8 \hat{i}+14 \hat{j}+9 \hat{k})$

(d) $\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})$

[Online April 15, 2018]

Sol.

Q. Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{c}=\hat{j}-\hat{k}$ and a vector $\vec{b}$ be such that $\vec{a} \times \vec{b}=\vec{c}$ and $\vec{a} \cdot \vec{b}=3$. Then $|\vec{b}|$ equals?

(a) $\sqrt{\frac{11}{3}}$

(b) $\frac{\sqrt{11}}{3}$

(c) $\frac{11}{\sqrt{3}}$

(d) $\frac{11}{3}$

[Online April 16, 2018]

Sol.

Q. If $\vec{a}, \vec{b}$, and $\vec{c}$ are unit vectors such that $\vec{a}+2 \vec{b}+2 \overrightarrow{\mathrm{c}}=\overrightarrow{0}$, then $|\vec{a} \times \overrightarrow{\mathrm{c}}|$ is equal to

(a) $\frac{1}{4}$

(b) $\frac{\sqrt{15}}{4}$

(c) $\frac{15}{16}$

(d) $\frac{\sqrt{15}}{16}$

[Online April 15, 2018]

Sol.

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