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

Mar 11, 2015

Hello,

• In a orthonormal basis, the distance between $A \left(x , y\right)$ and $A ' \left(x ' , y '\right)$ is

$d = \sqrt{{\left(x - x '\right)}^{2} + {\left(y - y '\right)}^{2}}$.

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

$x = r \cos \theta , y = r \sin \theta$
$x ' = r ' \cos \theta ' , y ' = r ' \sin \theta '$,

So,

$d = \sqrt{{\left(r \cos \theta - r ' \cos \theta '\right)}^{2} + {\left(r \sin \theta - r ' \sin \theta '\right)}^{2}}$

Develop, and use the formula ${\cos}^{2} x + {\sin}^{2} x = 1$. So you get :

$d = \sqrt{{r}^{2} - 2 r r ' \left(\cos \theta \cos \theta ' + \sin \theta \sin \theta '\right) + r {'}^{2}}$

Finally, you know that $\cos \theta \cos \theta ' + \sin \theta \sin \theta ' = \cos \left(\theta - \theta '\right)$, therefore,

$d = \sqrt{{r}^{2} + r {'}^{2} - 2 r r ' \cos \left(\theta - \theta '\right)}$.