Vector Algebra JEE Main PYQs (2020–2026)

Solved Vector Algebra PYQs

3 recent questions with answers. The full set of 213 is free to practise.

1. JEE Main 2026EasyAddition, Subtraction & Section Formula of Vectors
   Q1

   Let $O$ be the origin, $\overrightarrow{OP}=\overrightarrow{a}$ and $\overrightarrow{OQ}=\overrightarrow{b}$ . If $R$ is the point on $\overrightarrow{OP}$ such that $\overrightarrow{OP}=5\overrightarrow{OR}$ , and $M$ is the point such that $\overrightarrow{OQ}=5\overrightarrow{RM}$ , then $\overrightarrow{PM}$ is equal to :

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

   Show answer & solution

   Correct answer: (B)

   $$\begin{array}{rl}& \overrightarrow{OR}=\frac{\overrightarrow{OP}}{5}=\frac{\overrightarrow{a}}{5}\\ & \overrightarrow{RM}=\frac{\overrightarrow{OQ}}{5}=\frac{\overrightarrow{b}}{5}\\ & \overrightarrow{OM}-\overrightarrow{OR}=\frac{\overrightarrow{b}}{5}\\ & \overrightarrow{OM}=\frac{\overrightarrow{a}+\overline{b}}{5}\\ & \overrightarrow{PM}=\overrightarrow{OM}-\overrightarrow{OP}\\ & \frac{\overrightarrow{a}+\overrightarrow{b}}{5}-\overrightarrow{a}\\ & =\frac{\overrightarrow{b}-4\overrightarrow{a}}{5}\end{array}$$
2. JEE Main 2026EasyVector Equations & Solving Vector Relations
   Q2

   Let the vectors $\overrightarrow{a}=-\hat{i}+\hat{j}+3\hat{k}$ and $\overrightarrow{b}=\hat{i}+3\hat{j}+\hat{k}$ . For some $\lambda ,\mu \in \mathbb{R}$ , let $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$ .
   
   If $\overrightarrow{c}\cdot (3\hat{i}-6\hat{j}+2\hat{k})=10$ and $\overrightarrow{c}\cdot (\hat{i}+\hat{j}+\hat{k})=-2$ , then $|\overrightarrow{c}{|}^2$ is equal to :

   • A8
   • B12
   • C14
   • D15

   Show answer & solution

   Correct answer: (B)

   $$\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$$
   
   $$\overrightarrow{c}=(\mu -\lambda )\hat{i}+(\lambda +3\mu )\hat{j}+(3\lambda +\mu )\hat{k}$$
   
   $$\overrightarrow{c}\cdot (3\hat{i}-6\hat{j}+2\hat{k})=10$$
   
   $$3(\mu -\lambda )-6(\lambda +3\mu )+2(3\lambda +\mu )=10$$
   
   $$-3\lambda -13\mu =10\dots (1)$$
   
   $$\overrightarrow{c}\cdot (\hat{i}+\hat{j}+\hat{k})=-2$$
   
   $$5\mu +3\lambda =-2\dots (2)$$
   
   Solving (1) & (2)
   
   $$\lambda =1,\mu =-1$$
   
   $$\overrightarrow{c}=-2\hat{i}-2\hat{j}+2\hat{k}$$
   
   $$|\overrightarrow{c}{|}^2=12$$
3. JEE Main 2026MediumDot (Scalar) Product of Two Vectors & Projections
   Q3

   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 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$\frac{2\sqrt{3}}{5}$

   Show answer & solution

   Correct answer: (B)

   Let angle between $\overrightarrow{PQ}$ & $\overrightarrow{PS}$ is $\theta$
   
   $$\cos \theta =\frac{\overrightarrow{PQ}.\overrightarrow{PS}}{\left|\overrightarrow{PQ}\right|\left|\overrightarrow{PS}\right|}$$
   
   $$\cos \theta =\frac{0-1+0}{\sqrt{2}\sqrt{2}}$$
   
   $$\cos \theta =-\frac{1}{2}$$
   
   $$\theta =\frac{2\pi }{3}$$
   
   $$\alpha =\frac{2\pi }{3}-\frac{\pi }{2}\text{So}\alpha =\frac{\pi }{6}$$
   
   $$={\sin }^2\frac{5\alpha }{2}-{\sin }^2\frac{\alpha }{2}$$
   
   $$=sin\left(\frac{5\alpha }{2}+\frac{\alpha }{2}\right)\sin \left(\frac{5\alpha }{2}-\frac{\alpha }{2}\right)$$
   
   $$=sin\frac{6\alpha }{2}\times sin\frac{4\alpha }{2}$$
   
   $$=sin3\alpha \times sin2\alpha$$
   
   $$=sin\frac{\pi }{2}\times sin\frac{\pi }{3}$$
   
   $$=1\times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{2}$$

Topic-wise question count

Where JEE Main has asked from Vector Algebra, most frequent first.

• Dot (Scalar) Product of Two Vectors & Projections86
• Vector Equations & Solving Vector Relations46
• Cross (Vector) Product of Two Vectors & Area Applications30
• Scalar Triple Product & Coplanarity of Vectors24
• Addition, Subtraction & Section Formula of Vectors14
• Linear Combination, Linear Dependence & Independence8
• Vector Triple Product5

Year-wise Vector Algebra questions in JEE Main

Number of questions from this chapter in each year's paper(s).