1. The problem statement, all variables and given/known data This isn't exactly a "problem" per se , but I need to understand it for a course i'm taking. I'm trying to understand the significance and when to use the vector conversion matrices, or just the identities. I'll use an example that I made up, using rectangular to polar coordinates for simplicity so you can see what I'm not understanding. 2. Relevant equations I'm fairly comfortable with the following identities to convert polar to rectangular and vice verse : x = ρ cos Φ y = ρ sin Φ ρ = √(x2 + y2) Φ = tan-1 x/y but I've difficulty with: Ax = Aρ cos Φ - AΦ sinΦ Ay = Aρ sin Φ + AΦ cosΦ Aρ = Ax cos Φ + Ay sinΦ AΦ = -Ax sin Φ + Ay cosΦ In this example: (the lowercase scripts indicate unit vectors.) Apolar = 10 aρ + π/3 aΦ or 3. The attempt at a solution Using: x = ρ cos Φ, y = ρ sin Φ with ρ=10, Φ = π/3 ⇒ Arect = 5 ax + 5√3 ay which seems correct but using the matrix and pluging in for Aρ andAΦ Ax = Aρ cos Φ - AΦ sinΦ Ay = Aρ sin Φ + AΦ cosΦ i get: Ax = 10 cos π/3 - π/3 sin π/3 = 5 - π√3 / 6 ≈ 4 Ay = 10 sin π/3 + π/3 cos π/3 = 5√3 + π/6 ≈ 9.18 So I end up with a vector Arect with components : Arect = 4 ax + 9.18 ay So why don't both methods agree.