Vector Algebra Notes

Last reviewed 16 Sept 2026 · 8 min read

Scalars and vectors

Scalar — magnitude only (mass, length, time, work, energy, temperature).
Vector — magnitude and direction, obeying the parallelogram law (force, displacement, velocity, acceleration, moment).

Representation

a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}, ∣ a ∣ = a_{1}^{2} + a_{2}^{2} + a_{3}^{2}

Unit vector $\overset{a}{^} = \frac{a}{∣ a ∣}$ .
Position vector of point $P (x, y, z)$ : $r = x \hat{i} + y \hat{j} + z \hat{k}$ ; vector $A B = b - a$ .
Direction cosines $l = \frac{a _{1}}{∣ a ∣}$ , $m = \frac{a _{2}}{∣ a ∣}$ , $n = \frac{a _{3}}{∣ a ∣}$ with $l^{2} + m^{2} + n^{2} = 1$ ; direction ratios are proportional to $a_{1}, a_{2}, a_{3}$ .
Zero (null) vector, equal, collinear (parallel) and coplanar vectors.

Addition and section formula

Addition by triangle/parallelogram law; commutative and associative.
Section formula — point dividing AB internally in ratio $m : n$ :

r = \frac{m b + n a}{m + n}

(midpoint: $\frac{a + b}{2}$ ; centroid of a triangle: $\frac{a + b + c}{3}$ ).

Dot (scalar) product

FormulaDot product

a \cdot b = ∣ a ∣∣ b ∣ cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Angle: $cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$
Scalar projection of $a$ on $b$ : $\frac{a \cdot b}{∣ b ∣}$ ; vector projection: $(\frac{a \cdot b}{∣ b ∣ ^{2}}) b$
Work done by force $F$ through displacement $d$ : $W = F \cdot d$

Properties: commutative; distributive; $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ ; $\hat{i} \cdot \hat{j} = 0$ etc.; $a \cdot a = ∣ a ∣^{2}$ ; $a ⊥ b ⟺ a \cdot b = 0$ .

Cross (vector) product

FormulaCross product

a \times b = ∣ a ∣∣ b ∣ sin θ \overset{n}{^} = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}

$\overset{n}{^}$ is perpendicular to both vectors (right-hand rule).

Area of parallelogram with adjacent sides $a$ , $b$ : $∣ a \times b ∣$ ; area of triangle: $\frac{1}{2} ∣ a \times b ∣$
Moment (torque) of force $F$ acting at point with position $r$ about the origin: $M = r \times F$
Angular velocity: $v = ω \times r$
Lagrange identity: $∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$

Properties: anti-commutative $a \times b = - b \times a$ ; distributive; not associative; $\hat{i} \times \hat{j} = \hat{k}$ , $\hat{j} \times \hat{k} = \hat{i}$ , $\hat{k} \times \hat{i} = \hat{j}$ ; $\hat{i} \times \hat{i} = 0$ ; $a ∥ b ⟺ a \times b = 0$ .

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Vector Algebra