Vector Algebra formulas for JEE Main & Advanced

22 formula entries for Vector Algebra, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Position Vector

\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}

When to use. To locate a point in space relative to a fixed origin.

Trap. Confusing position vector with displacement vector (displacement is between two points).

Direction Cosines and Direction Ratios

l = \cos α, m = \cos β, n = \cos γ

l^{2} + m^{2} + n^{2} = 1

When to use. To describe the direction of a vector or line without magnitude.

Trap. Thinking direction ratios are unique; they are only proportional (e.g., $a, b, c$ and $k a, k b, k c$ are same).

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}

When to use. Combine vectors like forces, velocities, or displacements.

Trap. Adding magnitudes directly instead of components.

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}

When to use. To change the magnitude of a vector or to represent a vector in opposite direction.

Trap. Assuming $k$ must be positive; negative $k$ reverses direction.

Unit Vector

\hat{a} = \frac{\vec{a}}{∣ \vec{a} ∣}

When to use. To represent direction without magnitude, e.g., in force or velocity components.

Trap. Forgetting that $\hat{a}$ is dimensionless and has magnitude exactly 1.

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}

When to use. To find angle between vectors, projection, or work done.

Trap. Thinking dot product is a vector; it is a scalar.

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} ∣

When to use. To find area, torque, or a vector perpendicular to two vectors.

Trap. Forgetting that $\vec{a} \times \vec{b}$ is perpendicular to both $\vec{a}$ and $\vec{b}$ .

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} ∣

When to use. To check coplanarity of vectors or find volume.

Trap. Thinking it is a vector; it is a scalar.

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}

When to use. To simplify expressions involving multiple cross products.

Trap. Confusing the order: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ .

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}

When to use. To find coordinates of a point dividing a segment in a given ratio.

Trap. Confusing internal and external formulas; external uses subtraction.

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}

When to use. To find component of a vector along another direction.

Trap. Confusing scalar projection (a scalar) with vector projection (a vector).

Angle Between Two Vectors

\cos θ = \frac{\vec{a} \cdot \vec{b}}{∣ \vec{a} ∣ ∣ \vec{b} ∣}

When to use. To find angle between any two vectors or lines.

Trap. Forgetting that $θ$ is between 0 and $π$ .

Condition for Coplanarity of Three Vectors

\vec{a} \cdot (\vec{b} \times \vec{c}) = 0

When to use. To check if three vectors lie in the same plane.

Trap. Thinking two vectors are always coplanar; any two are coplanar.

Area of Triangle and Parallelogram

Area of parallelogram:

∣ \vec{a} \times \vec{b} ∣

; triangle:

\frac{1}{2} ∣ \vec{a} \times \vec{b} ∣

When to use. To compute area given two sides as vectors.

Trap. Using dot product instead of cross product.

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}

When to use. To simplify vector expressions and solve equations.

Trap. Assuming cross product is commutative.

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.

When to use. To check if vectors span a space or are redundant.

Trap. Thinking two vectors are always independent; they are dependent if collinear.

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}]}

When to use. In advanced vector analysis and crystallography; rarely in JEE but appears in some problems.

Trap. Thinking reciprocal vectors are just inverses; they are defined for a set of three non-coplanar vectors.

Using Vectors to Prove Geometry Theorems

Collinearity:

\vec{A B} = λ \vec{A C}

; Concurrency: solve for scalars.

When to use. To prove geometric properties like midpoints, centroids, or orthocenter.

Trap. Forgetting to set up origin appropriately.

Vector Equation of a Line

\vec{r} = \vec{a} + λ \vec{b}

When to use. To represent a line in 3D space.

Trap. Confusing $\vec{a}$ as a direction vector; it is a point on the line.

Vector Equation of a Plane

\vec{r} \cdot \vec{n} = d

(non-unit normal) or

\vec{r} \cdot \hat{n} = p

(distance from origin).

When to use. To represent a plane in 3D and find distances.

Trap. Confusing $\vec{n}$ with a point on the plane; it is a normal vector.

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} ∣}

When to use. To find minimum distance between two non-intersecting lines.

Trap. Using dot product instead of cross product in denominator.

Angle Between Two Planes

\cos θ = \frac{∣ {\vec{n}}_{1} \cdot {\vec{n}}_{2} ∣}{∣ {\vec{n}}_{1} ∣ ∣ {\vec{n}}_{2} ∣}

When to use. To find angle between two planes.

Trap. Using direction vectors of lines in the plane instead of normals.

Read the full chapter: Vector Algebra for JEE, with what JEE asks and the common traps.