How much of JEE Main comes from Vector Algebra?

JEEGenie estimates about 3% of JEE Main marks come from Vector Algebra, based on recent papers. The share varies from year to year.

Which formulas matter most in Vector Algebra?

The Vector Algebra sheet on this page has 22 formula entries, starting with Position Vector, Direction Cosines and Direction Ratios, Vector Addition and Subtraction, Multiplication of a Vector by a Scalar, Unit Vector. Each one comes with when to use it and the trap to avoid.

What topics does Vector Algebra cover for JEE?

Vector Algebra covers Vector Operations, Scalar (Dot) Product, Vector (Cross) Product, Scalar Triple Product, Vector Triple Product, Applications of Vectors.

How should I revise Vector Algebra before the exam?

Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Vector Algebra questions on JEEGenie.

How many previous-year questions are there on Vector Algebra?

JEEGenie has 350 previous-year questions on Vector Algebra, and this page shows a few with worked explanations.

Is practice on Vector Algebra free on JEEGenie?

Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.

Vector Algebra for JEE Main & Advanced — formulas, what is asked, and traps

JEEGenie estimates about 3% of JEE Main marks come from Vector Algebra. This page covers 22 formula entries, each with when to use it, what JEE asks and the traps. A common slip: Confusing a vector with its magnitude: e.g., $\vec{a}$ vs $∣ \vec{a} ∣$ .

Key points to know

Vector and Scalar. A vector has both magnitude and direction; a scalar has only magnitude.
Position Vector. Position vector of a point P is $\vec{O P}$ from origin O to P.
Direction Cosines and Direction Ratios. Direction cosines are cosines of angles with axes; direction ratios are proportional to them.
Unit Vector. A unit vector has magnitude 1 and indicates direction.
Properties of Dot and Cross Products. Dot product is commutative; cross product is anti-commutative. Both are distributive over addition.
Linear Combination and Linear Dependence. Vectors are linearly dependent if one can be expressed as combination of others; else independent.
Reciprocal System of Vectors. Three vectors form a reciprocal system if dot products with corresponding original vectors are 1 and others 0.
Vector Equation of a Line. A line can be expressed as $\vec{r} = \vec{a} + λ \vec{b}$ where $\vec{a}$ is a point and $\vec{b}$ is direction.
Vector Equation of a Plane. A plane can be expressed as $\vec{r} \cdot \hat{n} = d$ where $\hat{n}$ is unit normal.

Formula sheet

Name	Formula	When to use
Position Vector	$\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$	To locate a point in space relative to a fixed origin.
Direction Cosines and Direction Ratios	$l = \cos α, m = \cos β, n = \cos γ$ , $l^{2} + m^{2} + n^{2} = 1$	To describe the direction of a vector or line without magnitude.
Vector Addition and Subtraction	$\vec{a} + \vec{b} = (a_{x} + b_{x}) \hat{i} + (a_{y} + b_{y}) \hat{j} + (a_{z} + b_{z}) \hat{k}$	Combine vectors like forces, velocities, or displacements.
Multiplication of a Vector by a Scalar	$k \vec{a} = (k a_{x}) \hat{i} + (k a_{y}) \hat{j} + (k a_{z}) \hat{k}$	To change the magnitude of a vector or to represent a vector in opposite direction.
Unit Vector	$\hat{a} = \frac{\vec{a}}{∣ \vec{a} ∣}$	To represent direction without magnitude, e.g., in force or velocity components.
Dot Product (Scalar Product)	$\vec{a} \cdot \vec{b} = ∣ \vec{a} ∣ ∣ \vec{b} ∣ \cos θ = a_{x} b_{x} + a_{y} b_{y} + a_{z} b_{z}$	To find angle between vectors, projection, or work done.
Cross Product (Vector Product)	$\vec{a} \times \vec{b} = ∣ \vec{a} ∣ ∣ \vec{b} ∣ \sin θ \hat{n} = ∣ \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \end{array} ∣$	To find area, torque, or a vector perpendicular to two vectors.
Scalar Triple Product	$[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c}) = ∣ \begin{array}{ccc} a_{x} & a_{y} & a_{z} \\ b_{x} & b_{y} & b_{z} \\ c_{x} & c_{y} & c_{z} \end{array} ∣$	To check coplanarity of vectors or find volume.
Vector Triple Product	$\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c}$	To simplify expressions involving multiple cross products.
Section Formula for Vectors	For internal division: $\vec{r} = \frac{m \vec{b} + n \vec{a}}{m + n}$ ; external: $\vec{r} = \frac{m \vec{b} - n \vec{a}}{m - n}$	To find coordinates of a point dividing a segment in a given ratio.
Projection of a Vector	Scalar projection: $\frac{\vec{a} \cdot \vec{b}}{∣ \vec{b} ∣}$ ; Vector projection: $(\frac{\vec{a} \cdot \vec{b}}{∣ \vec{b} ∣^{2}}) \vec{b}$	To find component of a vector along another direction.
Angle Between Two Vectors	$\cos θ = \frac{\vec{a} \cdot \vec{b}}{∣ \vec{a} ∣ ∣ \vec{b} ∣}$	To find angle between any two vectors or lines.
Condition for Coplanarity of Three Vectors	$\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$	To check if three vectors lie in the same plane.
Area of Triangle and Parallelogram	Area of parallelogram: $∣ \vec{a} \times \vec{b} ∣$ ; triangle: $\frac{1}{2} ∣ \vec{a} \times \vec{b} ∣$	To compute area given two sides as vectors.
Properties of Dot and Cross Products	$\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ ; $\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}$	To simplify vector expressions and solve equations.
Linear Combination and Linear Dependence	$c_{1} \vec{a} + c_{2} \vec{b} + c_{3} \vec{c} = 0$ with not all $c_{i} = 0$ implies dependence.	To check if vectors span a space or are redundant.
Reciprocal System of Vectors	$\vec{a} \cdot {\vec{a}}^{'} = 1$ , $\vec{a} \cdot {\vec{b}}^{'} = 0$ , etc. where ${\vec{a}}^{'} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}$	In advanced vector analysis and crystallography; rarely in JEE but appears in some problems.
Using Vectors to Prove Geometry Theorems	Collinearity: $\vec{A B} = λ \vec{A C}$ ; Concurrency: solve for scalars.	To prove geometric properties like midpoints, centroids, or orthocenter.
Vector Equation of a Line	$\vec{r} = \vec{a} + λ \vec{b}$	To represent a line in 3D space.
Vector Equation of a Plane	$\vec{r} \cdot \vec{n} = d$ (non-unit normal) or $\vec{r} \cdot \hat{n} = p$ (distance from origin).	To represent a plane in 3D and find distances.
Shortest Distance Between Two Lines	$d = \frac{∣ ({\vec{a}}_{2} - {\vec{a}}_{1}) \cdot ({\vec{b}}_{1} \times {\vec{b}}_{2}) ∣}{∣ {\vec{b}}_{1} \times {\vec{b}}_{2} ∣}$	To find minimum distance between two non-intersecting lines.
Angle Between Two Planes	$\cos θ = \frac{∣ {\vec{n}}_{1} \cdot {\vec{n}}_{2} ∣}{∣ {\vec{n}}_{1} ∣ ∣ {\vec{n}}_{2} ∣}$	To find angle between two planes.

Open the Vector Algebra formula page

What JEE asks

Vector and Scalar. JEE often asks to classify quantities or to spot that a given statement mixes vector and scalar operations incorrectly.
Position Vector. JEE uses position vectors to find midpoints, centroids, and section formulas.
Direction Cosines and Direction Ratios. JEE asks to find direction cosines from direction ratios or vice versa, often in 3D geometry.
Vector Addition and Subtraction. JEE tests through problems on resultant forces or relative velocities.
Multiplication of a Vector by a Scalar. JEE uses this in problems involving unit vectors and linear combinations.
Unit Vector. JEE often asks to find a unit vector along a given vector or to express a vector in terms of its unit vector.
Dot Product (Scalar Product). JEE tests through problems on orthogonality, projection, and work-energy.
Cross Product (Vector Product). JEE asks to compute cross product, area of triangle/parallelogram, or torque.
Scalar Triple Product. JEE tests coplanarity conditions and volume calculations.
Vector Triple Product. JEE uses this to prove identities or solve geometry problems.
Section Formula for Vectors. JEE often asks for centroid, incenter, or points dividing medians.
Projection of a Vector. JEE tests through problems on work done (projection of force along displacement).
Angle Between Two Vectors. JEE asks to find angle between lines, planes, or vectors in geometry problems.
Condition for Coplanarity of Three Vectors. JEE tests through problems on points being coplanar or vectors being linearly dependent.
Area of Triangle and Parallelogram. JEE asks to find area of triangle or parallelogram in coordinate geometry or vector problems.
Properties of Dot and Cross Products. JEE tests these properties in identity proofs and problem solving.
Linear Combination and Linear Dependence. JEE asks to determine if given vectors are linearly independent or to express one as combination of others.
Reciprocal System of Vectors. JEE may ask to find reciprocal vectors or use them to solve equations.
Using Vectors to Prove Geometry Theorems. JEE often asks vector proofs for properties of triangles and quadrilaterals.
Vector Equation of a Line. JEE uses this to find intersection of lines or shortest distance.
Vector Equation of a Plane. JEE asks to find equation of plane given conditions or angle between planes.
Shortest Distance Between Two Lines. JEE tests through problems on skew lines and shortest distance.
Angle Between Two Planes. JEE asks to find angle between planes or to check if planes are perpendicular/parallel.

Common traps

Confusing a vector with its magnitude: e.g., $\vec{a}$ vs $∣ \vec{a} ∣$ . Vector and Scalar
Confusing position vector with displacement vector (displacement is between two points). Position Vector
Thinking direction ratios are unique; they are only proportional (e.g., $a, b, c$ and $k a, k b, k c$ are same). Direction Cosines and Direction Ratios
Adding magnitudes directly instead of components. Vector Addition and Subtraction
Assuming $k$ must be positive; negative $k$ reverses direction. Multiplication of a Vector by a Scalar
Forgetting that $\hat{a}$ is dimensionless and has magnitude exactly 1. Unit Vector
Thinking dot product is a vector; it is a scalar. Dot Product (Scalar Product)
Forgetting that $\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ . Cross Product (Vector Product)
Thinking it is a vector; it is a scalar. Scalar Triple Product
Confusing the order: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ . Vector Triple Product
Confusing internal and external formulas; external uses subtraction. Section Formula for Vectors
Confusing scalar projection (a scalar) with vector projection (a vector). Projection of a Vector
Forgetting that $θ$ is between 0 and $π$ . Angle Between Two Vectors
Thinking two vectors are always coplanar; any two are coplanar. Condition for Coplanarity of Three Vectors

Topics in Vector Algebra

Vector Operations
Scalar (Dot) Product
Vector (Cross) Product
Scalar Triple Product
Vector Triple Product
Applications of Vectors

Previous-year questions on Vector Algebra

JEEGenie has 350 previous-year questions on Vector Algebra. A few worked examples follow.

Year	Questions
2026	27
2025	26
2024	38
2023	53
2022	29
2021	44
2020	18
2019	18
2018	6
2017	6
2016	5
2015	4
2014	7
2013	6
2012	9
2011	4
2010	2
2009	1
2008	2
2007	2
2006	2
2005	5
2004	5
2003	9
2002	6
2001	4
1995	4
1993	1
1988	1
1987	1
1986	1
1984	2
1983	1
1981	1

Question 1 medium

\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 :

Answer: C

Explanation. Let

\vec{x} = 3 \vec{a} + 2 \vec{b}, \vec{y} = 3 \vec{a} - 2 \vec{b} .

Then

\vec{x} + \vec{y} = 6 \vec{a}, \vec{x} - \vec{y} = 4 \vec{b} .

∣ \vec{x} + \vec{y} ∣ = 6 ∣ \vec{a} ∣ = 12, ∣ \vec{x} - \vec{y} ∣ = 4 ∣ \vec{b} ∣ = 12.

We need to maximize

3 ∣ \vec{x} ∣ + 4 ∣ \vec{y} ∣ .

By the parallelogram law,

∣ \vec{x} ∣^{2} + ∣ \vec{y} ∣^{2} = \frac{∣ \vec{x} + \vec{y} ∣^{2} + ∣ \vec{x} - \vec{y} ∣^{2}}{2} = \frac{12^{2} + 12^{2}}{2} = 144.

Thus

∣ \vec{x} ∣^{2} + ∣ \vec{y} ∣^{2} = 144.

By Cauchy–Schwarz,

(3 ∣ \vec{x} ∣ + 4 ∣ \vec{y} ∣)^{2} \leq (3^{2} + 4^{2}) (∣ \vec{x} ∣^{2} + ∣ \vec{y} ∣^{2}) = 25 \cdot 144 = 3600.

Hence

3 ∣ \vec{x} ∣ + 4 ∣ \vec{y} ∣ \leq 60.

Equality occurs when

\frac{∣ \vec{x} ∣}{3} = \frac{∣ \vec{y} ∣}{4}

and the vectors

\vec{x}, \vec{y}

are chosen appropriately, so the maximum is attainable.

60

**Answer: C) 60**

From NTA/IIT papers: JEE Main 2026 · 2 Apr Shift 1

Question 2 easy

Let

∣ \vec{a} ∣ = 2, ∣ \vec{b} ∣ = 3

and the angle between the vectors

\vec{a}

and

\vec{b}

\frac{π}{4}

. Then

∣ (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) ∣^{2}

is equal to :

Answer: D

Explanation.

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

Since

\vec{a} \times \vec{a} = 0

and

\vec{b} \times \vec{b} = 0

= - 3 (\vec{a} \times \vec{b}) + 4 (\vec{b} \times \vec{a}) = - 3 (\vec{a} \times \vec{b}) - 4 (\vec{a} \times \vec{b}) = - 7 (\vec{a} \times \vec{b})

Thus,

∣ (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) ∣ = 7 ∣ \vec{a} \times \vec{b} ∣

Now,

∣ \vec{a} \times \vec{b} ∣ = ∣ \vec{a} ∣ ∣ \vec{b} ∣ \sin \frac{π}{4} = 2 \cdot 3 \cdot \frac{\sqrt{2}}{2} = 3 \sqrt{2}

So,

∣ (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) ∣ = 7 \cdot 3 \sqrt{2} = 21 \sqrt{2}

Squaring,

∣ (\vec{a} + 2 \vec{b}) \times (2 \vec{a} - 3 \vec{b}) ∣^{2} = (21 \sqrt{2})^{2} = 441 \cdot 2 = 882

D

From NTA/IIT papers: JEE Main 2023 · 13 Apr Shift 2

Frequently asked questions

How much of JEE Main comes from Vector Algebra?: JEEGenie estimates about 3% of JEE Main marks come from Vector Algebra, based on recent papers. The share varies from year to year.
Which formulas matter most in Vector Algebra?: The Vector Algebra sheet on this page has 22 formula entries, starting with Position Vector, Direction Cosines and Direction Ratios, Vector Addition and Subtraction, Multiplication of a Vector by a Scalar, Unit Vector. Each one comes with when to use it and the trap to avoid.
What topics does Vector Algebra cover for JEE?: Vector Algebra covers Vector Operations, Scalar (Dot) Product, Vector (Cross) Product, Scalar Triple Product, Vector Triple Product, Applications of Vectors.
How should I revise Vector Algebra before the exam?: Learn each formula together with its condition of use, read the trap listed beside it, then practise questions and review every mistake. The practice link on this page opens Vector Algebra questions on JEEGenie.
How many previous-year questions are there on Vector Algebra?: JEEGenie has 350 previous-year questions on Vector Algebra, and this page shows a few with worked explanations.
Is practice on Vector Algebra free on JEEGenie?: Sign-up is free and includes a 7-day Pro trial. After the trial, free accounts keep a limited daily quota.