Vector Algebra PYQs - Last 10 Years

JEE Main / Mathematics / Algebra / 238 recent questions

MathematicsAlgebra2017-2026

All Vector Algebra PYQs Last 5 Years Create chapter test

Practice 238 JEE Main Mathematics questions from Vector Algebra. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

238

PYQs on Page

Mathematics / Algebra

2017-2026

Year Range

Based on indexed question metadata

161

Last 5 Years

2022-2026

238

Last 10 Years

2017-2026

Recent Year Trend

2021

2022

2023

2024

2025

2026

202148 max PYQs/year2026

Question Types

238PYQs

MCQ76.5%

INTEGER23.5%

Difficulty Mix

#1 Medium203

#2 Hard21

#3 Easy14

161 in last 5 years238 in last 10 years

Last 10 Years Vector Algebra Questions

Showing 50 of 238 filtered questions.

1Vector Algebra

Let $\overrightarrow{\mathrm{a}}=4 \hat{i}-\hat{j}+3 \hat{k}, \overrightarrow{\mathrm{~b}}=10 \hat{i}+2 \hat{j}-\hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such that $2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b...

2Vector Algebra

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 $\_\_\_\_$ .

INTEGER+4 / -12026

3Vector Algebra

Let $\overrightarrow{\mathrm{a}}=2 \hat{i}+3 \hat{j}+3 \hat{k}$ and $\overrightarrow{\mathrm{b}}=6 \hat{i}+3 \hat{j}+3 \hat{k}$. Then the square of the area of the triangle with adjacent sides determined by the vectors $(2 \vec{a}+3 \vec{b}...

4Vector Algebra

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 $\vec{r} \times \vec{a}+\vec{a} \times \vec{b}=\overrightarrow{0}$ and $\vec{r} \cdot \vec{a}=0$, then $|3 \vec{r}|^2$ is equ...

5Vector Algebra

Let $O$ be the origin, $\overrightarrow{O P}=\vec{a}$ and $\overrightarrow{O Q}=\vec{b}$. If $R$ is the point on $\overrightarrow{O P}$ such that $\overrightarrow{O P}=5 \overrightarrow{O R}$, and $M$ is the point such that $\overrightarrow...

6Vector Algebra

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 $\overrightarrow{\mathrm{A}}=\lambda \hat{u}+\hat{v}+(\hat{u} \times \hat{v})$, then $\lambda$ is equal to:

7Vector Algebra

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}|=2$ and $|\vec{b}|=3$, then the maximum value of $3|(3 \vec{a}+2 \vec{b})|+4|(3 \vec{a}-2 \vec{b})|$ is :

8Vector Algebra

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{PQ} = \hat{j} + \hat{k}$ and $\overrightarrow{PS} = \hat{i} - \hat{j}$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the para...

9Vector Algebra

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})...

10Vector Algebra

Let $P Q R$ be a triangle such that $\overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $\overrightarrow{\mathrm{PR}}=a \hat{\mathrm{i}}+b \hat{\mathrm{j}}-4 \hat{\mathrm{k}}, a, b \in \mathrm{Z}$. Let S be the point on QR , which is eq...

INTEGER+4 / -12026

11Vector Algebra

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying
\(|\vec{a}-\vec{b}|^2+|\vec{b}-\vec{c}|^2+|\vec{c}-\vec{a}|^2=9 \text { and }|2 \vec{a}+k \vec{b}+k \vec{c}|=3 \text {, }\)
the positive value of k is :

12Vector Algebra

If the distance of the point $P(43, \alpha, \beta)$, $\beta < 0$, from the line $\vec{r} = 4\hat{i} - \hat{k} + \mu (2\hat{i} + 3\hat{k}), \mu \in \mathbb{R}$ along a line with direction ratios $3, -1, 0$ is $13\sqrt{10}$, then $\alpha^2 + ...

INTEGER+4 / -12026

13Vector Algebra

Let P be a point in the plane of the vectors $\overrightarrow{AB}=3\hat{i} + \hat{j} - \hat{k}$ and $\overrightarrow{AC}=\hat{i} - \hat{j} + 3\hat{k}$ such that P is equidistant from the lines AB and AC. If $|\overrightarrow{AP}| = \frac{\s...

14Vector Algebra

Let the lines $\mathrm{L}_1: \vec{r}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \lambda \in \mathbb{R}$ and $\mathrm{L}_2: \vec{r}=(4 \hat{\mathrm{i}}+\hat{\math...

15Vector Algebra

Let $\vec{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\vec{c}=\vec{a} \times \vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=...

16Vector Algebra

Let $\vec{a}=2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\vec{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. If $\vec{c}$ is a vector such that $2(\vec{a} \times \vec{c})+3(\vec{b} \times \vec{c})=\overrightarr...

17Vector Algebra

Let $\vec{a}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\vec{c}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. Let $\vec{v}$ be the vector in the plane ...

18Vector Algebra

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\hat{i}-\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{d}...

19Vector Algebra

Let $\vec{a}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \vec{b}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{c}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\vec{v}=\vec{a} \times \vec{b}$. If ...

20Vector Algebra

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 $60^{\circ}$, then $|\vec{a} \cdot \ve...

21Vector Algebra

Let $\overrightarrow{\mathrm{AB}}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on the diagona...

22Vector Algebra

Let a vector $\overrightarrow{\mathrm{a}}=\sqrt{2} \hat{i}-\hat{j}+\lambda \hat{k}, \lambda>0$, make an obtuse angle with the vector $\overrightarrow{\mathrm{b}}=-\lambda^2 \hat{i}+4 \sqrt{2} \hat{j}+4 \sqrt{2} \hat{k}$ and an angle $\theta...

INTEGER+4 / -12026

23Vector Algebra

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2 \hat{k}, \lambda \in \boldsymbol{Z}$ be two vectors. Let $\vec{c}=\vec{a} \times \vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in $y z$-plane. If $|\vec{c}|=\sqrt{53...

24Vector Algebra

Let $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ be vectors such that $|\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and $\overrightarrow{\mathrm{c}} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}...

25Vector Algebra

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+2 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=8 \hat{i}+7 \hat{j}-3 \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c...

26Vector Algebra

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the line $\overrightarrow{\mathrm{r}}=(-\hat{i}+3 \hat{j}+\hat{k})+\lambda(2 \hat{i}+3 \hat{j}-\hat{k})$.
Then the length o...

27Vector Algebra

For a triangle $A B C$, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and $\vec{r}=\overrightarrow{B A}$. If $|\vec{p}|=2 \sqrt{3},|\vec{q}|=2$ and $\cos \theta=\frac{1}{\sqrt{3}}$, where $\theta$ is the angle between $\v...

28Vector Algebra

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 ...

29Vector Algebra

Let the angle $\theta, 0<\theta<\frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin ^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c}=3 \hat{a}+6 \hat{b}+9(\hat{a} \times \hat{b})$, then the value of $9(\ve...

30Vector Algebra

Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that
$ \frac{|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|}{|\vec{a} + \vec{b}| - |\vec{a} - \vec{b}|} = \sqrt{2} + 1. $ Then $ \frac{|\vec{a} + \vec{b}|^2}{|\vec{a}|^2} ...

31Vector Algebra

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 $\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)$.
Let $\vec{v}=\vec{v}_1+\overrightarrow{v_2...

32Vector Algebra

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 $A B C$. Then $6\left(|\overrightarrow{\math...

INTEGER+4 / -12025

33Vector Algebra

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...

INTEGER+4 / -12025

34Vector Algebra

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 $|(\...

INTEGER+4 / -12025

35Vector Algebra

If $\overrightarrow{\mathrm{a}}$ is a 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 $\overrightarrow{\mathrm{a}}$ is :

36Vector Algebra

Let $\overrightarrow{\mathrm{a}}=2 \hat{i}-3 \hat{j}+\hat{k}, \quad \overrightarrow{\mathrm{~b}}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and a vector $\overrightarrow{\mathrm{c}}$ be such that $(\vec{a}-\vec{c}) \times \vec{b}=-18 \hat{i}-3 \hat{j}+...

37Vector Algebra

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{a} \times \vec{b} = \vec{c} \times \vec{b} $ and $ (\vec{a} + \vec{c}) \cdot (\...

38Vector Algebra

Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+7 \hat{j}+3 \hat{k}$. Let $\mathrm{L}_1 : \overrightarrow{\mathrm{r}}=(-\hat{i}+2 \hat{j}+\hat{k})+\lambda \vec{a}, \mathrm{\lambda} \in \mathbf{R}$ and $\mathrm{L}_2: \overrigh...

39Vector Algebra

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 $ \hat{a} $ makes an angle of $ \cos^{-1} \left( -\frac{1}{3} \right) $ with the ...

40Vector Algebra

Let $\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}$. If $\overrigh...

INTEGER+4 / -12025

41Vector Algebra

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...

42Vector Algebra

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...

43Vector Algebra

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} \cd...

44Vector Algebra

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 $\...

45Vector Algebra

Let $\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{k}$. Then the project...

46Vector Algebra

Let the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and C of a tetrahedron ABCD be $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-2 \hat{k}$ and $2 \hat{i}+\hat{j}-\hat{k}$ respectiv...

47Vector Algebra

Let the arc $A C$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc $A C$ such that $\frac{\text { length of } \operatorname{arc} A B}{\text { length of } \operatorname{arc} B C}=\frac{1...

48Vector Algebra

Let the point A divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r: 1(r>0)$. If O is the origin and $(\overrightarrow{\mathrm{OQ}} \cdot \overrightarrow{\mathrm{OA}})-\frac{1...

49Vector Algebra

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and ...

INTEGER+4 / -12025

50Vector Algebra

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. 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 $...