Question

What's the difference in finding the distance between two polar coordinates and two rectangular coordinate?

Answer 1

Hello,

• In a orthonormal basis, the distance between `A(x,y)` and `A'(x',y')` is

`d = sqrt((x-x')^2 + (y-y')^2)`.

• With polar coordinates, `A[t, theta]` and `A'[r',theta']`, you have to write the relations :

`x = r cos theta, y = r sin theta`
`x' = r' cos theta', y' = r' sin theta'`,

So,

`d = sqrt((r cos theta - r' cos theta')^2 + (r sin theta - r' sin theta')^2 )`

Develop, and use the formula `cos^2 x + sin^2 x = 1`. So you get :

`d = sqrt(r^2 - 2 rr' (cos theta cos theta' + sin theta sin theta')+ r'^2)`

Finally, you know that `cos theta cos theta' + sin theta sin theta' = cos(theta - theta')`, therefore,

`d = sqrt(r^2 + r'^2 - 2rr' cos(theta - theta'))`.