- #1

FrankJ777

- 140

- 6

## Homework Statement

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.

## Homework 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 = √(x

Φ = tan

y = ρ sin Φ

ρ = √(x

^{2}+ y^{2})Φ = tan

^{-1}x/ybut I've difficulty with:

**A**

A

A

A

_{x}= A_{ρ}cos Φ - A_{Φ }sinΦA

_{y}= A_{ρ}sin Φ + A_{Φ }cosΦA

_{ρ}= A_{x}cos Φ + A_{y }sinΦA

_{Φ}= -A_{x}sin Φ + A_{y }cosΦIn this example: (the lowercase scripts indicate unit vectors.)

**A**or

_{polar}= 10 a_{ρ}+ π/3 a_{Φ}## The Attempt at a Solution

Using:

**x = ρ cos Φ, y = ρ sin Φ**

with ρ=10, Φ = π/3

⇒with ρ=10, Φ = π/3

⇒

**A**_{rect}= 5 a_{x}+ 5√3 a_{y}which seems correct

but using the matrix and pluging in for

**A**and

_{ρ}**A**

_{Φ }**A**

A

_{x}=**A**cos Φ - A_{ρ}_{Φ }sinΦA

_{y}=**A**sin Φ + A_{ρ}_{Φ }cosΦi get:

A

A

A

_{x}=**10**cos**π/3**- π/3 sin**π/3 = 5 - π√3 / 6 ≈ 4**A

_{y}= 10 sin**π/3**+**π/3**cos**π/3 = 5√3 + π/6 ≈ 9.18**So I end up with a vector A

_{rect}with components :

**A**

_{rect}= 4 a_{x}+ 9.18 a_{y}So why don't both methods agree.

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