Vector Algebra - JEE Main Previous Year Question with Solutions
Vector Algebra JEE Main previous year questions span topics including scalar triple product, cross product, dot product, collinear vectors, and projection of vectors. Between 1–2 questions appear in every JEE Main session. Practising these questions with solutions helps you identify patterns, avoid sign errors, and consistently score full marks in this chapter.
JEE Main Previous Year Question of Math with Solutions are available at eSaral. Practicing JEE Main Previous Year Papers Questions of mathematics will help the JEE aspirants in realizing the question pattern as well as help in analyzing weak & strong areas.eSaral helps the students in clearing and understanding each topic in a better way. eSaral is providing complete chapter-wise notes of Class 11th and 12th both for all subjects.Besides this, eSaral also offers NCERT Solutions, Previous year questions for JEE Main and Advanced, Practice questions, Test Series for JEE Main, JEE Advanced and NEET, Important questions of Physics, Chemistry, Math, and Biology and many more.Download eSaral app for free study material and video tutorials.
Q. If $\overrightarrow{\mathrm{u}}, \overrightarrow{\mathrm{v}}, \overrightarrow{\mathrm{w}}$ are non-coplanar vectors and $\mathrm{p}, \mathrm{q}$ are real numbers, then the equality $[3 \vec{u} p \vec{v} p \vec{w}]-[p \vec{v} \vec{w} q \vec{u}]-[2 \vec{w} q \vec{v} q \vec{u}]=0$ holds for $:-$ (1) More than two but not all values of (p,q)(2) All values of (p, q)(3) Exactly one value of (p, q)(4) Exactly two values of (p, q)[AIEEE-2009]
Ans. (3)
Q. Let $\vec{a}=\hat{j}-\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k} .$ Then the vector $\vec{b}$ satisfying $\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}$ and $\vec{a} . \vec{b}=3$ is : (1) $-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$(2) $2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$(3) $\hat{\mathrm{i}}-\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$(4) $\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$[AIEEE-2010]
Q. The vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ are not perpendicular and $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{d}}$ are two vectors satisfying: $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{d}}$ and $\overrightarrow{\mathrm{a} .} \overrightarrow{\mathrm{d}}=0 .$ Then the vector $\overrightarrow{\mathrm{d}}$ is equal to :- $(1) \overrightarrow{\mathrm{b}}+\left(\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{c}}$(2) $\overrightarrow{\mathrm{c}}-\left(\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{b}}$(3) $\overrightarrow{\mathrm{b}}-\left(\frac{\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{c}}$(4) $\overrightarrow{\mathrm{c}}+\left(\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}}{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}}\right) \overrightarrow{\mathrm{b}}$ [AIEEE-2011]
Ans. (2)
Q. If $\overrightarrow{\mathrm{a}}=\frac{1}{\sqrt{10}}(3 \hat{\mathrm{i}}+\hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{b}}=\frac{1}{7}(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}),$ then the value of $(2 \overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}) \cdot[(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}})]$ is 1: (1) 5 (2) 3 (3) – 5 (4) – 3[AIEEE-2011]
Ans. (3) $=-4 a^{2}-b^{2}+4 \vec{a} \cdot \vec{b}=-5$
Q. Let $\vec{a}, \vec{b}, \vec{c}$ be three non-zero vectors which are pairwise non-collinear. If $\vec{a}+3 \vec{b}$ is collinear with $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{c}}$ is colliner with $\overrightarrow{\mathrm{a}},$ then $\overrightarrow{\mathrm{a}}+3 \overrightarrow{\mathrm{b}}+6 \overrightarrow{\mathrm{c}}$ is: (1) $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}$ ( 2)$\quad \vec{a}$ ( 3)$\quad \overrightarrow{\mathrm{c}}$ ( 4) $\overrightarrow{0}$[AIEEE-2011]
Ans. (4)
Q. Let $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ be two unit vectors. If the vectors $\overrightarrow{\mathbf{c}}=\hat{\mathbf{a}}+2 \hat{\mathbf{b}}$ and $\overrightarrow{\mathbf{d}}=5 \hat{\mathbf{a}}-4 \hat{\mathbf{b}}$ are perpendicular to each other, then the angle between $\hat{\mathbf{a}}$ and $\hat{\mathbf{b}}$ is : (1) $\frac{\pi}{4}$( 2)$\frac{\pi}{6}$(3) $\frac{\pi}{2}$( 4)$\frac{\pi}{3}$[AIEEE-2012]
Q. Let ABCD be a parallelogram such that $\overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{q}}, \overrightarrow{\mathrm{AD}}=\overrightarrow{\mathrm{p}}$ and $\angle \mathrm{BAD}$ be an acute angle. If $\overrightarrow{\mathrm{r}}$ is the vector that coincides with the altitude directed from the vertex $B$ to the side $A D,$ then $\overrightarrow{\mathrm{r}}$ is given by : [AIEEE-2012]
Ans. (3)
Q. If the vectors $\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=5 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ are the sides of a triangle ABC, then the length of the median through A is : (1) $\sqrt{18}$(2) $\sqrt{72}$(3) $\sqrt{33}$(4) $\sqrt{45}$[JEE-MAINS 2013]
Ans. (3) $\overrightarrow{\mathrm{AD}}=\frac{\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}}{2}=4 \hat{\mathrm{j}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}$$|\overrightarrow{\mathrm{AD}}|=\sqrt{33}$
Q. Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ be three vectors. A vectors. A vectors of the type $\overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{c}}$ for some scalar $\lambda,$ whose projection on $\overrightarrow{\mathrm{a}}$ is of magnitude $\sqrt{\frac{2}{3}},$ is : (1) $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$(2) $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$(3) $2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}}$(4) $2 \hat{i}+3 \hat{j}+3 \hat{k}$[JEE-MAINS Online 2013]
Q. Let $\vec{a}=2 \hat{i}+\hat{j}-2 \hat{k}, \vec{b}=\hat{i}+\hat{j} .$ If $\vec{c}$ is a vector such that $\vec{a} \bullet \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle between $\vec{a} \times \vec{b}$ and $\vec{c}$ is $30^{\circ},$ then $|(\vec{a} \times \vec{b}) \times \vec{c}|$ equals : ( 1)$\frac{3}{2}$(2) 3( 3)$\frac{1}{2}$(4) $\frac{3 \sqrt{3}}{2}$[JEE-MAINS Online 2013]
Q. Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero vectors such that no two of them are collinear and $(\vec{a}, \times \vec{b}) \times \vec{c}=\frac{1}{3}|\vec{b}||\vec{c}| \vec{a} \cdot$ If $\theta$ is the angle between vectors $\vec{b}$ and $\vec{c},$ then a value of $\sin \theta$ is : [JEE(Main)-2015]
Q. Let $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$ be three unit vectors such that $\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\frac{\sqrt{3}}{2}(\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) .$ If $\overrightarrow{\mathrm{b}}$ is not parallel to $\overrightarrow{\mathrm{c}}$ then the angle between a and $\overrightarrow{\mathrm{b}}$ is :- (1) $\frac{5 \pi}{6}$ (2) $\frac{3 \pi}{4}$ (3) $\frac{\pi}{2}$ (4) $\frac{2 \pi}{3}$[JEE(Main)-2016]
Q. Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}} .$ Let $\overrightarrow{\mathrm{c}}$ be a vector such that $|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=3,|(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}|=3$ and the angle between $\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ be $30^{\circ} .$ Then $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}$ is equal to : ( 1)$\frac{1}{8}$ (2) $\frac{25}{8}$ (3) 2 (4) 5[JEE(Main)-2018]
Q. If the position vectors of the vertices $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ of a $\Delta \mathrm{ABC}$ 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 $\angle \mathrm{A}$ meets $\mathrm{BC}$ is : (1) $\frac{1}{2}(4 \hat{i}+8 \hat{j}+11 \hat{k})$(2) $\frac{1}{3}(6 \hat{i}+13 \hat{j}+18 \hat{k})$(3) $\frac{1}{4}(8 \hat{\mathrm{i}}+14 \hat{\mathrm{j}}+19 \hat{\mathrm{k}})$(4) $\frac{1}{3}(6 \hat{i}+11 \hat{j}+15 \hat{k})$[JEE(Main)-2018]
Ans. (2)
Q. If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}},$ and $\overrightarrow{\mathrm{c}}$ are unit vectors such that $\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}+2 \overrightarrow{\mathrm{c}}=\overrightarrow{0},$ then $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}|$ is equal to : [JEE(Main)-2018]
Ans. (4)
Q. If the angle between the lines, $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{5-x}{-2}=\frac{7 y-14}{p}=\frac{z-3}{4}$ is $\cos ^{-1}\left(\frac{2}{3}\right),$ then $p$ is equal to: [JEE(Main)-2018]
Ans. (2)
Q. Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{c}}=\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and a vector $\overrightarrow{\mathrm{b}}$ be such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=3 .$ Then $|\overrightarrow{\mathrm{b}}|$ equals : (1) $\sqrt{\frac{11}{3}}$(2) $\frac{11}{\sqrt{3}}$(2) $\frac{11}{\sqrt{3}}$( 4)$\frac{11}{3}$[JEE(Main)-2018]
Ans. (1)
Q. Let $\vec{u}$ be a vector coplanar with the vectors $\vec{a}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\vec{b}=\hat{j}+\hat{k} .$ If $\vec{u}$ is perpendicular to $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{u} . \overrightarrow{\mathrm{b}}}=24,$ then $|\overrightarrow{\mathrm{u}}|^{2}$ is equal to- (1) 315 (2) 256 (3) 84 (4) 336[JEE(Main)-2018]
Ans. (4)
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Frequently Asked Questions
Find answers to common questions.
Is Vector Algebra difficult for JEE Main?
Vector Algebra is considered moderate in difficulty. The formula set is finite and well-defined, which means questions follow recognisable patterns. The main challenge is execution under time pressure — specifically, sign handling in triple products and correct application of the BAC-CAB identity. Solving 40–50 PYQs systematically makes this chapter very manageable.
What are the most important topics in Vector Algebra for JEE Main?
The most important topics are scalar triple product, vector triple product (BAC-CAB rule), cross product and its applications, projection of one vector on another, and conditions for collinearity and coplanarity. Among these, scalar triple product and the BAC-CAB identity together account for roughly 60% of all Vector Algebra PYQs
How many questions come from Vector Algebra in JEE Main?
Vector Algebra contributes 1–2 questions per JEE Main session, as confirmed by NTA's official paper pattern across 2020–2024. Each question carries 4 marks, so you can realistically target 4–8 marks from this chapter every attempt. Given the short formula list and repeating question types, this is one of the most reliable chapters for consistent scoring
Where can I find complete NCERT solutions for Vector Algebra Class 12?
You can find complete worked solutions for all NCERT Class 12 Maths Chapter 10 (Vector Algebra) exercises at eSaral's NCERT Solutions for Class 12 Maths. These solutions follow the same methodology used in JEE solutions, making them directly useful for exam preparation rather than just board-level understanding.
What is the BAC-CAB rule and how is it used in JEE Main?
The BAC-CAB rule states: a × (b × c) = (a·c)b − (a·b)c. In JEE Main, it is used to simplify vector triple product expressions into scalar dot product terms, which can then be evaluated using given conditions like perpendicularity or known magnitudes. A common question type gives you a triple product equal to a scalar multiple of a vector and asks you to find the angle between two of the vectors.
How is scalar triple product tested in JEE Main?
Scalar triple product is tested in two main ways: (1) finding the value of a parameter for which vectors are coplanar, i.e., [a b c] = 0; and (2) simplifying an expression involving [a × b, b × c, c × a] using the identity that this equals [a b c]². Both question types appear regularly and can be solved in under two minutes with practice.
$=-4 a^{2}-b^{2}+4 \vec{a} \cdot \vec{b}=-5$ 
[AIEEE-2012] 
$\overrightarrow{\mathrm{AD}}=\frac{\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AC}}}{2}=4 \hat{\mathrm{j}}-\hat{\mathrm{j}}+4 \hat{\mathrm{k}}$ $|\overrightarrow{\mathrm{AD}}|=\sqrt{33}$ 
[JEE(Main)-2015] 
[JEE(Main)-2018] 
[JEE(Main)-2018]