Mathematics · JEE

Vector Algebra Formula Sheet for JEE

82+ JEE formulas in this unit

Quick answer

The Vector Algebra JEE formula sheet lists 82+ important formulas for JEE Main and Advanced, including essential identities from Vectors and scalars, the addition of vectors, Components of a vector in two dimensions and three-dimensional spaces, Scalar and vector products. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Vector Algebra. This unit-wise formula list covers 82+ exam-relevant results across Vectors and scalars, the addition of vectors, Components of a vector in two dimensions and three-dimensional spaces, Scalar and vector products, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

82 formulas across 3 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 47Important: 32Supplementary: 3

Vectors and scalars, the addition of vectors

Scalar and vector quantitiesSupplementary

\text{Scalars have only magnitude; vectors have magnitude and direction.}

Where used in JEE: Basic classification of quantities in vector algebra problems.

Equality of vectorsImportant

\vec{a}=\vec{b} \iff |\vec{a}|=|\vec{b}|\text{ and they have the same direction}

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Checking vector equality in geometric and algebraic questions.

Negative of a vectorImportant

|-\vec{a}|=|\vec{a}|,\quad -\vec{a}\text{ is opposite in direction to }\vec{a}

Variables: $\vec{a}$ is a vector.
Where used in JEE: Vector subtraction, resolving direction changes.

Zero vectorImportant

|\vec{0}|=0,\quad \vec{a}+\vec{0}=\vec{a}

Variables: $\vec{0}$ is the zero vector; $\vec{a}$ is any vector.
Where used in JEE: Identity element in vector addition.

Unit vector along a vectorEssential

\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}

Variables: $\hat{a}$ is the unit vector in the direction of $\vec{a}$ .
Conditions: $\vec{a}\neq \vec{0}$ .
Where used in JEE: Finding direction of a vector, projection, components.

Magnitude of scalar multipleImportant

|\lambda \vec{a}|=|\lambda|\,|\vec{a}|

Variables: $\lambda$ is a scalar, $\vec{a}$ is a vector.
Where used in JEE: Scaling vectors, comparing lengths after multiplication.

Direction under scalar multiplicationSupplementary

\lambda\vec{a}

has the same direction as

\vec{a}

\lambda>0

, opposite direction if

\lambda<0

Variables: $\lambda$ is a scalar, $\vec{a}$ is a vector.
Where used in JEE: Geometric interpretation of scalar multiplication.

Vector addition triangle lawImportant

\vec{a}+\vec{b}

is represented by the vector from the tail of

\vec{a}

to the head of

\vec{b}

when

\vec{b}

is placed at the head of

\vec{a}

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Geometric addition of two vectors.

Vector addition parallelogram lawImportant

\vec{a}+\vec{b}

is the diagonal of the parallelogram formed by

\vec{a}

and

\vec{b}

from the common initial point.

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Geometric addition, resultant of two vectors.

Commutative law of vector additionEssential

\vec{a}+\vec{b}=\vec{b}+\vec{a}

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Simplifying vector expressions.

Associative law of vector additionEssential

(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Manipulating sums of multiple vectors.

Distributive law of scalar multiplication over vector additionEssential

\lambda(\vec{a}+\vec{b})=\lambda\vec{a}+\lambda\vec{b}

Variables: $\lambda$ is a scalar; $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Expanding vector expressions.

Distributive law over scalar additionImportant

(\lambda+\mu)\vec{a}=\lambda\vec{a}+\mu\vec{a}

Variables: $\lambda,\mu$ are scalars; $\vec{a}$ is a vector.
Where used in JEE: Combining like vector terms.

Associative law of scalar multiplicationImportant

\lambda(\mu\vec{a})=(\lambda\mu)\vec{a}

Variables: $\lambda,\mu$ are scalars; $\vec{a}$ is a vector.
Where used in JEE: Simplifying scalar multiples of vectors.

Vector subtractionEssential

\vec{a}-\vec{b}=\vec{a}+(-\vec{b})

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Difference of vectors, displacement between points.

Section formula in vector formEssential

\overrightarrow{OP}=\dfrac{m\overrightarrow{OB}+n\overrightarrow{OA}}{m+n}

Variables: $P$ divides $AB$ internally in the ratio $AP:PB=m:n$ ; $O$ is the origin.
Conditions: $m,n>0$ for internal division, $m+n\neq 0$ .
Where used in JEE: Finding position vector of a point dividing a line segment.

External section formula in vector formImportant

\overrightarrow{OP}=\dfrac{m\overrightarrow{OB}-n\overrightarrow{OA}}{m-n}

Variables: $P$ divides $AB$ externally in the ratio $AP:PB=m:n$ ; $O$ is the origin.
Conditions: $m\neq n$ .
Where used in JEE: External division of a line segment in vector form.

Position vector of midpointEssential

\overrightarrow{OM}=\dfrac{\overrightarrow{OA}+\overrightarrow{OB}}{2}

Variables: $M$ is the midpoint of $AB$ ; $O$ is the origin.
Where used in JEE: Midpoint problems, centroid and section formula applications.

Centroid of triangle in vector formImportant

\overrightarrow{OG}=\dfrac{\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}}{3}

Variables: $G$ is the centroid of triangle $ABC$ ; $O$ is the origin.
Where used in JEE: Centroid and median problems in coordinate/vector geometry.

General linear combination of vectorsImportant

\vec{r}=x\vec{a}+y\vec{b}+z\vec{c}

Variables: $x,y,z$ are scalars; $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Expressing a vector in terms of given vectors; basis representation.

Collinearity in vector formImportant

\overrightarrow{AB}=\lambda\overrightarrow{AC}

Variables: $\lambda$ is a scalar.
Conditions: Points $A,B,C$ are collinear iff one displacement vector is a scalar multiple of the other.
Where used in JEE: Testing whether points are collinear.

Components of a vector in two dimensions and three-dimensional spaces

Standard unit vectors in 2D and 3DEssential

\hat{i},\hat{j},\hat{k}

are unit vectors along the positive

x

y

-, and

z

-axes respectively.

Where used in JEE: Resolving vectors into rectangular components.

Representation of vector in 2DEssential

\vec{a}=a_x\hat{i}+a_y\hat{j}

Variables: $a_x,a_y$ are components of $\vec{a}$ along $x$ - and $y$ -axes.
Where used in JEE: Coordinate form of vectors in a plane.

Representation of vector in 3DEssential

\vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}

Variables: $a_x,a_y,a_z$ are components of $\vec{a}$ along $x$ -, $y$ -, and $z$ -axes.
Where used in JEE: Coordinate form of vectors in space.

Component form using ordered triplesImportant

\vec{a}\equiv (a_x,a_y,a_z)

Variables: $a_x,a_y,a_z$ are rectangular components of $\vec{a}$ .
Conditions: In 2D, use $(a_x,a_y)$ .
Where used in JEE: Compact notation in vector calculations.

Equality of vectors in component formImportant

a_x=b_x,\ a_y=b_y,\ a_z=b_z \iff \vec{a}=\vec{b}

Variables: $\vec{a}=(a_x,a_y,a_z),\ \vec{b}=(b_x,b_y,b_z)$ .
Conditions: Use only relevant coordinates in 2D.
Where used in JEE: Comparing vectors using components.

Addition of vectors in component formEssential

(a_x,a_y,a_z)+(b_x,b_y,b_z)=(a_x+b_x,\ a_y+b_y,\ a_z+b_z)

Variables: $(a_x,a_y,a_z)$ and $(b_x,b_y,b_z)$ are vectors.
Conditions: In 2D, omit the $z$ -component.
Where used in JEE: Direct vector addition in coordinates.

Scalar multiplication in component formEssential

\lambda(a_x,a_y,a_z)=(\lambda a_x,\lambda a_y,\lambda a_z)

Variables: $\lambda$ is a scalar.
Conditions: In 2D, omit the $z$ -component.
Where used in JEE: Scaling coordinate vectors.

Magnitude of a 2D vectorEssential

|\vec{a}|=\sqrt{a_x^2+a_y^2}

Variables: $\vec{a}=a_x\hat{i}+a_y\hat{j}$ .
Where used in JEE: Length of a vector in a plane.

Magnitude of a 3D vectorEssential

|\vec{a}|=\sqrt{a_x^2+a_y^2+a_z^2}

Variables: $\vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$ .
Where used in JEE: Length of a vector in space.

Direction cosines of a vectorEssential

l=\dfrac{a_x}{|\vec{a}|},\quad m=\dfrac{a_y}{|\vec{a}|},\quad n=\dfrac{a_z}{|\vec{a}|}

Variables: $l,m,n$ are direction cosines of $\vec{a}=a_x\hat{i}+a_y\hat{j}+a_z\hat{k}$ .
Conditions: $\vec{a}\neq \vec{0}$ .
Where used in JEE: Angles with coordinate axes, unit direction vectors.

Relation among direction cosinesEssential

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

Variables: $l,m,n$ are direction cosines of a vector.
Where used in JEE: Checking validity of direction cosines, finding unknown cosine.

Vector from one point to another in 2D/3DEssential

\overrightarrow{AB}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}

Variables: $A(x_1,y_1,z_1),\ B(x_2,y_2,z_2)$ .
Conditions: In 2D, take $z_1=z_2=0$ or omit the $z$ -term.
Where used in JEE: Displacement vector between two points.

Distance between two pointsImportant

AB=|\overrightarrow{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

Variables: $A(x_1,y_1,z_1),\ B(x_2,y_2,z_2)$ .
Conditions: In 2D, omit the $z$ -term.
Where used in JEE: Length and geometry problems using vector components.

Position vector of a pointEssential

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

Variables: $(x,y,z)$ are coordinates of the point relative to the origin.
Conditions: In 2D, $\vec{r}=x\hat{i}+y\hat{j}$ .
Where used in JEE: Coordinate-vector conversion.

Unit vector in terms of direction cosinesImportant

\hat{a}=l\hat{i}+m\hat{j}+n\hat{k}

Variables: $l,m,n$ are the direction cosines of $\vec{a}$ .
Conditions: $l^2+m^2+n^2=1$ .
Where used in JEE: Constructing a unit vector from directional information.

Vector with given magnitude and direction cosinesImportant

\vec{a}=|\vec{a}|(l\hat{i}+m\hat{j}+n\hat{k})

Variables: $|\vec{a}|$ is magnitude; $l,m,n$ are direction cosines.
Conditions: $l^2+m^2+n^2=1$ .
Where used in JEE: Building a vector from its magnitude and direction.

Component of a vector along axes using anglesEssential

\vec{a}=|\vec{a}|(\cos\alpha\,\hat{i}+\cos\beta\,\hat{j}+\cos\gamma\,\hat{k})

Variables: $\alpha,\beta,\gamma$ are angles made by $\vec{a}$ with positive $x$ -, $y$ -, and $z$ -axes.
Conditions: $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$ .
Where used in JEE: Problems involving angles a vector makes with axes.

Direction ratios and direction cosines relationImportant

l:m:n=a:b:c

, \quad

l=\dfrac{a}{\sqrt{a^2+b^2+c^2}},\ m=\dfrac{b}{\sqrt{a^2+b^2+c^2}},\ n=\dfrac{c}{\sqrt{a^2+b^2+c^2}}

Variables: $a,b,c$ are direction ratios; $l,m,n$ are direction cosines.
Conditions: Not all of $a,b,c$ are zero.
Where used in JEE: Converting DRs to DCs in spatial direction problems.

Resolved part of a vector along coordinate axes in 2DEssential

a_x=|\vec{a}|\cos\theta,\quad a_y=|\vec{a}|\sin\theta

Variables: $\theta$ is the angle made by $\vec{a}$ with the positive $x$ -axis.
Where used in JEE: Resolving planar vectors into rectangular components.

Scalar and vector products

Scalar (dot) product definitionEssential

\vec{a}\cdot\vec{b}=|\vec{a}|\,|\vec{b}|\cos\theta

Variables: $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ , $0\le \theta\le \pi$ .
Conditions: For nonzero vectors when using angle form.
Where used in JEE: Angle between vectors, projections, orthogonality, work-like quantities.

Dot product in component formEssential

\vec{a}\cdot\vec{b}=a_xb_x+a_yb_y+a_zb_z

Variables: $\vec{a}=(a_x,a_y,a_z),\ \vec{b}=(b_x,b_y,b_z)$ .
Conditions: In 2D, omit the $z$ -term.
Where used in JEE: Direct computation of scalar product from coordinates.

Dot products of unit vectorsEssential

\hat{i}\cdot\hat{i}=\hat{j}\cdot\hat{j}=\hat{k}\cdot\hat{k}=1,\quad \hat{i}\cdot\hat{j}=\hat{j}\cdot\hat{k}=\hat{k}\cdot\hat{i}=0

Conditions: For mutually perpendicular standard unit vectors.
Where used in JEE: Expanding dot products in component form.

Angle between two vectorsEssential

\cos\theta=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}|\,|\vec{b}|}

Variables: $\theta$ is the angle between nonzero vectors $\vec{a},\vec{b}$ .
Conditions: $\vec{a}\neq\vec{0},\ \vec{b}\neq\vec{0}$ .
Where used in JEE: Finding acute/obtuse/right angle between vectors.

Orthogonality conditionEssential

\vec{a}\cdot\vec{b}=0 \iff \vec{a}\perp\vec{b}

Variables: $\vec{a},\vec{b}$ are vectors.
Conditions: For nonzero vectors in the reverse implication; zero vector is orthogonal to every vector in dot-product sense.
Where used in JEE: Testing perpendicularity.

Self dot productEssential

\vec{a}\cdot\vec{a}=|\vec{a}|^2

Variables: $\vec{a}$ is a vector.
Where used in JEE: Magnitude calculations, simplifying expressions.

Projection of one vector on anotherEssential

\operatorname{proj}_{\vec{b}}\vec{a}=\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|^2}\right)\vec{b}

Variables: $\vec{a}$ is projected on $\vec{b}$ .
Conditions: $\vec{b}\neq\vec{0}$ .
Where used in JEE: Vector projection and decomposition problems.

Scalar component of a vector along another vectorEssential

\operatorname{comp}_{\vec{b}}\vec{a}=\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|}=|\vec{a}|\cos\theta

Variables: $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ .
Conditions: $\vec{b}\neq\vec{0}$ .
Where used in JEE: Resolved part of a vector along a given direction.

Work form of dot productSupplementary

W=\vec{F}\cdot\vec{s}=|\vec{F}|\,|\vec{s}|\cos\theta

Variables: $\vec{F}$ is force, $\vec{s}$ is displacement.
Where used in JEE: Applied problems involving scalar product.

Properties of dot productEssential

\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a},\quad \vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c},\quad (\lambda\vec{a})\cdot\vec{b}=\lambda(\vec{a}\cdot\vec{b})

Variables: $\lambda$ is a scalar; $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Algebraic manipulation of scalar products.

Magnitude of sum using dot productEssential

|\vec{a}+\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2+2\vec{a}\cdot\vec{b}

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Resultant magnitude, angle between vectors, geometry problems.

Magnitude of difference using dot productEssential

|\vec{a}-\vec{b}|^2=|\vec{a}|^2+|\vec{b}|^2-2\vec{a}\cdot\vec{b}

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Distance, diagonal, and angle-related problems.

Parallelogram law for magnitudesImportant

|\vec{a}+\vec{b}|^2+|\vec{a}-\vec{b}|^2=2(|\vec{a}|^2+|\vec{b}|^2)

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Standard identity in vector magnitude problems.

Condition for acute, obtuse, right angleImportant

\vec{a}\cdot\vec{b}>0\Rightarrow \theta<\dfrac{\pi}{2},\quad \vec{a}\cdot\vec{b}<0\Rightarrow \theta>\dfrac{\pi}{2},\quad \vec{a}\cdot\vec{b}=0\Rightarrow \theta=\dfrac{\pi}{2}

Variables: $\theta$ is the angle between nonzero vectors $\vec{a},\vec{b}$ .
Conditions: $\vec{a},\vec{b}\neq\vec{0}$ .
Where used in JEE: Determining nature of angle using dot product sign.

Cauchy-Schwarz inequalityImportant

|\vec{a}\cdot\vec{b}|\le |\vec{a}|\,|\vec{b}|

Variables: $\vec{a},\vec{b}$ are vectors.
Conditions: Equality iff $\vec{a}$ and $\vec{b}$ are parallel.
Where used in JEE: Bounds, proving inequalities, checking feasibility.

Triangle inequalityImportant

|\vec{a}+\vec{b}|\le |\vec{a}|+|\vec{b}|

Variables: $\vec{a},\vec{b}$ are vectors.
Conditions: Equality iff $\vec{a}$ and $\vec{b}$ are in the same direction.
Where used in JEE: Magnitude bounds and geometric interpretation.

Condition for parallel vectors using dot productEssential

\vec{a}\parallel\vec{b} \iff \vec{a}=\lambda\vec{b}

Variables: $\lambda$ is a scalar.
Conditions: For nonzero vectors.
Where used in JEE: Testing parallelism.

Vector (cross) product definitionEssential

|\vec{a}\times\vec{b}|=|\vec{a}|\,|\vec{b}|\sin\theta

Variables: $\theta$ is the angle between $\vec{a}$ and $\vec{b}$ .
Conditions: Direction is perpendicular to both $\vec{a}$ and $\vec{b}$ according to right-hand rule.
Where used in JEE: Area, perpendicular vectors, moment-like quantities.

Cross product vector formEssential

\vec{a}\times\vec{b}=|\vec{a}|\,|\vec{b}|\sin\theta\,\hat{n}

Variables: $\hat{n}$ is a unit vector perpendicular to the plane of $\vec{a},\vec{b}$ in the right-hand-rule direction.
Where used in JEE: Constructing the actual vector product.

Cross products of unit vectorsEssential

\hat{i}\times\hat{j}=\hat{k},\quad \hat{j}\times\hat{k}=\hat{i},\quad \hat{k}\times\hat{i}=\hat{j}

Conditions: Cyclic order.
Where used in JEE: Expanding cross products in component form.

Anti-cyclic cross productsEssential

\hat{j}\times\hat{i}=-\hat{k},\quad \hat{k}\times\hat{j}=-\hat{i},\quad \hat{i}\times\hat{k}=-\hat{j}

Where used in JEE: Sign handling in cross-product expansion.

Cross product of a vector with itself and zero vectorImportant

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

Variables: $\vec{a}$ is a vector.
Where used in JEE: Simplifying expressions and testing parallelism.

Anti-commutative law of cross productEssential

\vec{a}\times\vec{b}=-(\vec{b}\times\vec{a})

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Reversing order in vector products.

Distributive law of cross productEssential

\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Expanding vector product expressions.

Scalar multiplication with cross productImportant

(\lambda\vec{a})\times\vec{b}=\lambda(\vec{a}\times\vec{b})=\vec{a}\times(\lambda\vec{b})

Variables: $\lambda$ is a scalar.
Where used in JEE: Simplifying scaled vector products.

Cross product in determinant formEssential

\vec{a}\times\vec{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_x&a_y&a_z\\b_x&b_y&b_z\end{vmatrix}

Variables: $\vec{a}=(a_x,a_y,a_z),\ \vec{b}=(b_x,b_y,b_z)$ .
Where used in JEE: Computing cross product in coordinate form.

Cross product in expanded component formEssential

\vec{a}\times\vec{b}=(a_yb_z-a_zb_y)\hat{i}-(a_xb_z-a_zb_x)\hat{j}+(a_xb_y-a_yb_x)\hat{k}

Variables: $\vec{a}=(a_x,a_y,a_z),\ \vec{b}=(b_x,b_y,b_z)$ .
Where used in JEE: Direct component-wise computation of vector product.

Area of parallelogram using cross productEssential

\text{Area}=|\vec{a}\times\vec{b}|

Variables: $\vec{a},\vec{b}$ are adjacent side vectors.
Where used in JEE: Area problems in 3D geometry.

Area of triangle using cross productEssential

\text{Area}=\dfrac{1}{2}|\vec{a}\times\vec{b}|

Variables: $\vec{a},\vec{b}$ are two side vectors from the same vertex.
Where used in JEE: Triangle area in vector form.

Condition for parallel vectors using cross productEssential

\vec{a}\times\vec{b}=\vec{0} \iff \vec{a}\parallel\vec{b}

Variables: $\vec{a},\vec{b}$ are vectors.
Conditions: For nonzero vectors in the reverse implication.
Where used in JEE: Testing parallelism and collinearity.

Sine formula for angle between vectorsImportant

\sin\theta=\dfrac{|\vec{a}\times\vec{b}|}{|\vec{a}|\,|\vec{b}|}

Variables: $\theta$ is the angle between nonzero vectors $\vec{a},\vec{b}$ .
Conditions: $\vec{a},\vec{b}\neq\vec{0}$ .
Where used in JEE: Finding angle when cross product magnitude is known.

Relation between dot and cross productsImportant

|\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2=|\vec{a}|^2|\vec{b}|^2

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Mixed angle/magnitude problems and identities.

Scalar triple productEssential

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Volume, coplanarity, determinant-based vector problems.

Scalar triple product determinant formEssential

\vec{a}\cdot(\vec{b}\times\vec{c})=\begin{vmatrix}a_x&a_y&a_z\\b_x&b_y&b_z\\c_x&c_y&c_z\end{vmatrix}

Variables: $\vec{a}=(a_x,a_y,a_z),\ \vec{b}=(b_x,b_y,b_z),\ \vec{c}=(c_x,c_y,c_z)$ .
Where used in JEE: Computing scalar triple product from coordinates.

Volume of parallelepipedEssential

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

Variables: $\vec{a},\vec{b},\vec{c}$ are concurrent edge vectors.
Where used in JEE: 3D geometry and volume problems.

Volume of tetrahedronImportant

V=\dfrac{1}{6}|\vec{a}\cdot(\vec{b}\times\vec{c})|

Variables: $\vec{a},\vec{b},\vec{c}$ are three edges from the same vertex.
Where used in JEE: Volume of tetrahedron and coordinate geometry in space.

Cyclic property of scalar triple productImportant

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Rearranging triple products for easier evaluation.

Sign change under interchange in scalar triple productImportant

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Conditions: Interchanging any two vectors changes the sign.
Where used in JEE: Handling orientation and determinant sign.

Coplanarity conditionEssential

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Conditions: Equivalent to coplanarity of the three vectors.
Where used in JEE: Testing coplanarity of vectors or four points.

Vector triple product expansionImportant

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Lagrange's formula; simplifying nested cross products.

Alternative vector triple product expansionImportant

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

Variables: $\vec{a},\vec{b},\vec{c}$ are vectors.
Where used in JEE: Expanding cross product of a cross product.

Lagrange's identityImportant

|\vec{a}\times\vec{b}|^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2

Variables: $\vec{a},\vec{b}$ are vectors.
Where used in JEE: Area, angle, and identity-based simplifications.

Frequently asked questions

What are the important Vector Algebra formulas for JEE?

This page lists 82+ JEE-relevant Vector Algebra formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Vector Algebra formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Vector Algebra, covering Vectors and scalars, the addition of vectors, Components of a vector in two dimensions and three-dimensional spaces, Scalar and vector products.

How should I revise the Vector Algebra formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Vector Algebra MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Vector Algebra?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.