Vectors formulas

Every formula JEE Advanced expects you to know for Vectors, with the variables spelled out. Free, no sign-up. Read it before a mock, not during one.

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Formulas: 36
Subject: Mathematics
Exam: JEE Advanced
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Vectors, every formula

36 formulas, typeset and free. Print it, or keep it open beside your practice.

Magnitude of a vector
$∣ a ∣ = x^{2} + y^{2} + z^{2}, a = x \hat{i} + y \hat{j} + z \hat{k}$
Unit vector
$\overset{a}{^} = \frac{a}{∣ a ∣}$
Vector along AB
$A B = b - a$
position vectors a, b
Vector of magnitude m parallel to a
$\pm \frac{m a}{∣ a ∣}$
Section formula (internal)
$O C = \frac{m b + n a}{m + n}$
Section formula (external)
$O C = \frac{m b - n a}{m - n}$
Midpoint
$O C = \frac{a + b}{2}$
Centroid of a triangle
$G = \frac{a + b + c}{3}$
Triangle inequality
$∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$
Collinear vectors
$a ∥ b \Leftrightarrow a = m b$
Collinearity of 3 points
$x a + y b + z c = 0, x + y + z = 0 (not all zero)$
Coplanar vectors
$x a + y b + z c = 0 (not all zero)$
Linear independence (3 vectors)
$Δ = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} \neq = 0$
Dot product (definition)
$a \cdot b = ∣ a ∣∣ b ∣ cos θ$
Dot product (components)
$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Angle via dot product
$cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$
Perpendicular vectors
$a \cdot b = 0 \Leftrightarrow a ⊥ b$
Projection of b on a
$\frac{a \cdot b}{∣ a ∣}$
scalar projection
|a±b|²
$∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2 a \cdot b$
Work done
$W = F \cdot S$
Cross product (definition)
$a \times b = ∣ a ∣∣ b ∣ sin θ \overset{n}{^}$
Cross product (determinant)
$a \times b = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}$
Anticommutativity
$a \times b = - (b \times a)$
Parallel vectors (cross)
$a \times b = 0 \Leftrightarrow a ∥ b$
Unit normal to a and b
$\overset{n}{^} = \pm \frac{a \times b}{∣ a \times b ∣}$
Area of a triangle
$Area = \frac{1}{2} ∣ a \times b ∣$
adjacent sides a, b
Area of a parallelogram
$Area = ∣ a \times b ∣$
Parallelogram via diagonals
$Area = \frac{1}{2} ∣ d_{1} \times d_{2} ∣$
Scalar triple product
$[a b c] = a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$
Coplanarity (STP)
$[a b c] = 0 \Leftrightarrow coplanar$
Volume of parallelepiped
$V = [a b c]$
Volume of tetrahedron
$V = \frac{1}{6} [a b c]$
Vector triple product
$a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$
Lagrange's identity
$∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$
Vector eqn of a line
$r = a + t b$
through a, parallel to b
i,j,k products
$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$

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Vectors formulas

Vectors, every formula

Magnitude of a vector

Unit vector

Vector along AB

Vector of magnitude m parallel to a

Section formula (internal)

Section formula (external)

Midpoint

Centroid of a triangle

Triangle inequality

Collinear vectors

Collinearity of 3 points

Coplanar vectors

Linear independence (3 vectors)

Dot product (definition)

Dot product (components)

Angle via dot product

Perpendicular vectors

Projection of b on a

|a±b|²

Work done

Cross product (definition)

Cross product (determinant)

Anticommutativity

Parallel vectors (cross)

Unit normal to a and b

Area of a triangle

Area of a parallelogram

Parallelogram via diagonals

Scalar triple product

Coplanarity (STP)

Volume of parallelepiped

Volume of tetrahedron

Vector triple product

Lagrange's identity

Vector eqn of a line

i,j,k products

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