Math · Coordinate Geometry and Vectors

Vectors formulas for JEE

Every Vectors formula you need for JEE, grouped by concept.

FormulasRevision notes
29 formulas6 concepts
01Vectors Basics02Vector Operations & Section Formula03Scalar (Dot) Product & Projections04Vector (Cross) Product05Scalar Triple Product06Vector Triple Product
01

Vectors Basics

6 formulas

Direction cosines from ratios

l=aa2+b2+c2,m=ba2+b2+c2,n=ca2+b2+c2l = \frac{a}{\sqrt{a^2+b^2+c^2}}, m = \frac{b}{\sqrt{a^2+b^2+c^2}}, n = \frac{c}{\sqrt{a^2+b^2+c^2}}

Converting direction ratios (a, b, c) into direction cosines (l, m, n).

applies whena,b,ca, b, c not all zero.
direction_ratiosdirection_cosines

Direction cosines normalization

l2+m2+n2=1l^2+m^2+n^2=1

The sum of the squares of the direction cosines of a vector is always 1.

direction_cosines

Vector components in 3D

r⃗=xi^+yj^+zk^\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}

Expression of a position vector using orthogonal unit vectors.

componentsbasis

Vector magnitude

∣r⃗∣=x2+y2+z2|\vec{r}| = \sqrt{x^2 + y^2 + z^2}

Magnitude or length of a vector in 3D coordinate space.

magnitudelength

Vector joining two points

P1P2⃗=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^\vec{P_1P_2} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k}

Displacement vector directed from point P1 to P2.

pointsdisplacement

Unit vector

a^=a⃗∣a⃗∣\hat{a} = \frac{\vec{a}}{|\vec{a}|}

Vector of magnitude 1 pointing in the exact same direction as the given vector.

applies when∣a⃗∣≠0|\vec{a}| \neq 0
unit_vector
02

Vector Operations & Section Formula

4 formulas

Vector addition (components)

a⃗+b⃗=(a1+b1)i^+(a2+b2)j^+(a3+b3)k^\vec{a} + \vec{b} = (a_1+b_1)\hat{i} + (a_2+b_2)\hat{j} + (a_3+b_3)\hat{k}

Algebraic sum of two vectors given in rectangular component form.

additioncomponents

Section formula (External)

r⃗=mb⃗−na⃗m−n\vec{r} = \frac{m\vec{b} - n\vec{a}}{m-n}

Position vector of a point dividing a line segment externally in ratio m:n.

applies whenm≠nm \neq n
section_formulaexternal

Section formula (Internal)

r⃗=mb⃗+na⃗m+n\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}

Position vector of a point dividing a line segment internally in ratio m:n.

applies whenm+n≠0m+n \neq 0
section_formulainternal

Triangle Law of Vector Addition

AB⃗+BC⃗=AC⃗\vec{AB} + \vec{BC} = \vec{AC}

Net displacement rule for geometric addition of two vectors.

applies whenVectors must be arranged head-to-tail.
triangle_lawaddition
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03

Scalar (Dot) Product & Projections

7 formulas

Angle from dot product

cos⁡θ=a⃗⋅b⃗∣a⃗∣∣b⃗∣\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}

Used to find the angle theta between two non-zero vectors.

applies when∣a⃗∣≠0,∣b⃗∣≠0|\vec{a}| \neq 0, |\vec{b}| \neq 0
angledot_product

Cauchy-Schwarz Inequality

∣a⃗⋅b⃗∣≤∣a⃗∣∣b⃗∣|\vec{a} \cdot \vec{b}| \leq |\vec{a}||\vec{b}|

The absolute value of the dot product is bounded by the product of vector magnitudes.

inequalitycauchy_schwarz

Dot product

a⃗⋅b⃗=a1b1+a2b2+a3b3\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3

Algebraic scalar product computed using rectangular components.

dot_productcomponents

Dot product definition

a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta

Geometric definition of the scalar product.

applies whenBoth vectors are non-zero.
dot_productscalar_product

Scalar projection

p=a⃗⋅b⃗∣b⃗∣p = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}

Magnitude of the projection of vector a onto vector b.

applies when∣b⃗∣≠0|\vec{b}| \neq 0
projectionscalar

Vector projection

p⃗=(a⃗⋅b⃗∣b⃗∣2)b⃗\vec{p} = \left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\right)\vec{b}

Vector representing the projection of vector a onto vector b.

applies when∣b⃗∣≠0|\vec{b}| \neq 0
projectionvector

Triangle Inequality

∣a⃗+b⃗∣≤∣a⃗∣+∣b⃗∣|\vec{a} + \vec{b}| \leq |\vec{a}| + |\vec{b}|

The magnitude of a vector sum is bounded by the sum of individual magnitudes.

inequalitytriangle
04

Vector (Cross) Product

7 formulas

Angle from cross product

sin⁡θ=∣a⃗×b⃗∣∣a⃗∣∣b⃗∣\sin\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}||\vec{b}|}

Sine of the angle between two vectors.

applies when∣a⃗∣≠0,∣b⃗∣≠0|\vec{a}| \neq 0, |\vec{b}| \neq 0
anglecross_product

Area of parallelogram

A=∣a⃗×b⃗∣A = |\vec{a} \times \vec{b}|

Area of a parallelogram with adjacent sides formed by vectors a and b.

applies whenVectors must be coinitial adjacent sides.
areaparallelogramcross_product

Area of parallelogram (diagonals)

A=12∣d1⃗×d2⃗∣A = \frac{1}{2}|\vec{d_1} \times \vec{d_2}|

Area of a parallelogram using its diagonal vectors.

applies whend1 and d2 are diagonal vectors.
areaparallelogramcross_productjee-advanced

Area of triangle

Δ=12∣a⃗×b⃗∣\Delta = \frac{1}{2}|\vec{a} \times \vec{b}|

Area of a triangle with adjacent sides formed by vectors a and b.

applies whenVectors must be coinitial adjacent sides.
areatrianglecross_product

Cross product (components)

a⃗×b⃗=∣i^j^k^a1a2a3b1b2b3∣\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}

Vector product computed using a 3x3 determinant of vector components.

cross_productdeterminantcomponents

Cross product magnitude

a⃗×b⃗=∣a⃗∣∣b⃗∣sin⁡θn^\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta \hat{n}

Geometric definition of the vector product generating an orthogonal vector.

applies whenVectors are non-zero.
cross_productvector_product

Lagrange's Identity

∣a⃗×b⃗∣2+(a⃗⋅b⃗)2=∣a⃗∣2∣b⃗∣2|\vec{a} \times \vec{b}|^2 + (\vec{a} \cdot \vec{b})^2 = |\vec{a}|^2|\vec{b}|^2

Relates the squared magnitudes of the cross product and dot product.

identitylagrangejee-advanced
05

Scalar Triple Product

4 formulas

STP in component form

[a⃗b⃗c⃗]=∣a1a2a3b1b2b3c1c2c3∣[\vec{a} \vec{b} \vec{c}] = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}

Scalar triple product evaluated using a 3x3 determinant of their scalar components.

stpdeterminantjee-advanced

Scalar Triple Product

[a⃗b⃗c⃗]=a⃗⋅(b⃗×c⃗)[\vec{a} \vec{b} \vec{c}] = \vec{a} \cdot (\vec{b} \times \vec{c})

Scalar operation evaluating the dot product of one vector with the cross product of two others.

stptriple_productjee-advanced

Volume of parallelepiped

V=∣[a⃗b⃗c⃗]∣V = |[\vec{a} \vec{b} \vec{c}]|

Volume of a parallelepiped formed by three coterminous vectors.

applies whenVectors must represent coterminous edges.
volumeparallelepipedjee-advanced

Volume of tetrahedron

V=16∣[a⃗b⃗c⃗]∣V = \frac{1}{6}|[\vec{a} \vec{b} \vec{c}]|

Volume of a tetrahedron formed by three coterminous vectors.

applies whenVectors must represent coterminous edges.
volumetetrahedronjee-advanced
06

Vector Triple Product

1 formula

Vector Triple Product

a⃗×(b⃗×c⃗)=(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}

Expansion of the vector triple product, commonly referred to as the BAC-CAB rule.

vtptriple_productjee-advanced
Other chapters
MathSets34 formulasMathRelations and Functions18 formulasMathTrigonometric Functions56 formulasMathComplex Numbers and Quadratic Equations36 formulasMathPermutations and Combinations21 formulasMathBinomial Theorem24 formulasMathSequence and Series25 formulasMathStraight Lines27 formulasMathConic Sections35 formulasMathIntroduction to Three-dimensional Geometry9 formulasMathLimits and Derivatives32 formulasMathStatistics25 formulasMathProbability17 formulasMathRelations and Functions14 formulasMathInverse Trigonometric Functions33 formulasMathMatrices44 formulasMathDeterminants23 formulasMathContinuity and Differentiability30 formulasMathApplications of Derivatives26 formulasMathIntegrals47 formulasMathApplications of the Integrals14 formulasMathDifferential Equations23 formulasMathThree-dimensional Geometry32 formulasMathProbability15 formulas

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