Vector Algebra

A quantity that has magnitude as well as direction is called a vector.

Magnitude of position vector: |OP| = √(x² + y² + z²)

Magnitude of component-form vector: |r⃗| = |xî + yĵ + zk̂| = √(x² + y² + z²)

Zero vector: initial point = terminal point

Unit vector in direction of a⃗: â = (1/|a⃗|)a⃗

cos α = x/r, cos β = y/r, cos γ = z/r; l² + m² + n² = 1

Numbers lr, mr, nr are called direction ratios

Two vectors are equal iff corresponding components are equal

Two vectors are collinear iff there exists nonzero λ with b⃗ = λa⃗

Triangle law: AC⃗ = AB⃗ + BC⃗

Parallelogram law: resultant is diagonal through the common point

a⃗ + b⃗ = b⃗ + a⃗

(a⃗ + b⃗) + c⃗ = a⃗ + (b⃗ + c⃗)

a⃗ + 0⃗ = a⃗