Vector Algebra: General Concepts

Basics, Addition Laws & Section Formulas

Term	Definition / Formula
Magnitude (Modulus)	If $$ \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} $$, then: $$ \|\vec{r}\| = \sqrt{x^2 + y^2 + z^2} $$
Unit Vector (Direction only)	$$ \hat{a} = \frac{\vec{a}}{\|\vec{a}\|} $$ (Vector divided by its magnitude)
Collinear Vectors	Two vectors $$ \vec{a} $$ and $$ \vec{b} $$ are collinear if: $$ \vec{b} = \lambda \vec{a} $$ (for some scalar $$ \lambda $$)
Direction Cosines (l, m, n)	$$ l = \frac{x}{\|\vec{r}\|}, \quad m = \frac{y}{\|\vec{r}\|}, \quad n = \frac{z}{\|\vec{r}\|} $$

A. Triangle Law

If vectors are represented by two sides of a triangle in sequence:

$$ \vec{AB} + \vec{BC} = \vec{AC} $$

B. Parallelogram Law

If vectors act as adjacent sides of a parallelogram:

Diagonal $$ \vec{d} = \vec{a} + \vec{b} $$

*Properties: Vector addition is Commutative ($$ \vec{a}+\vec{b} = \vec{b}+\vec{a} $$) and Associative.

Points $$ P(x_1, y_1, z_1) $$ and $$ Q(x_2, y_2, z_2) $$	Vector $$ \vec{PQ} $$ (Position Vector of Q – P) $$ \vec{PQ} = (x_2-x_1)\hat{i} + (y_2-y_1)\hat{j} + (z_2-z_1)\hat{k} $$
Magnitude (Distance PQ): $$ \|\vec{PQ}\| = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} $$

Point $$ R(\vec{r}) $$ dividing the line joining $$ P(\vec{a}) $$ and $$ Q(\vec{b}) $$ in ratio $$ m:n $$

Internal Division
(R lies between P and Q)

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

External Division
(R lies outside P and Q)

$$ \vec{r} = \frac{m\vec{b} – n\vec{a}}{m – n} $$

Mid-Point Formula ($$ m:n = 1:1 $$)
$$ \vec{r} = \frac{\vec{a} + \vec{b}}{2} $$