Pre Calculus 6.3 - 6.5 formulas

component form

u =

Linear combination

u = Pi + qJ

Polar form

u = ||u||cosθi + ||u||sinθJ

magnitude formula

||u|| = √(P^2 + q^2)

unit vector

(P/||u||i) + (q/||u||J)

Reference angle

tan-1(q/P)

Dot product

u ⋅ v = (u1)(v1) + (u2)(v2)

c(u ⋅ v)

cu ⋅ v or u ⋅ cv

Angle between two vectors

cosθ = (u ⋅ v)/||u|| x ||v||

orthogonal vector

when u ⋅ v = 0 (perpendicular)

Trigonometric form

Z = a + bi

Multiplication of complex numbers

Z1×Z2 = r1×r2(cos(θ1 + θ2) + i sin(θ1 + θ2))

Division of complex numbers

Z1÷Z2 = r1÷r2(cos(θ1 - θ2) + sin(θ1 - θ2))

Exponents of complex numbers

Z^n = r^2(cosnθ + isinnθ)

root of complex numbers

u = r^(1/2)(cos(1/2)(θ +- 2πk) + isin(1/2)(θ +- 2πk))