Question

What is the formula for the distance between two polar coordinates?

Answer 1

`sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2)`

Explanation:

The distance is `sqrt(r_1^2+r_2^2-2r_1r_2cos(theta_1-theta_2)` if we are given `P_1=(r_1, theta_1)` and `P_2=(r_2, theta_2)`.

This is an application of the cosine law. Taking the difference between `theta_1` and `theta_2` gives us the angle between side `r_1` and side `r_2`. And the cosine law gives us the length of the `3^(rd)` side.

Answer 2

See below.

Explanation:

Given in cartesian coordinates.

`P_1=(x_1,y_1)` and `P_2= (x_2,y_2)`

the transition formulas

`{(x=r cos theta),(y=r sin theta):}`

then

`(x_1,y_1) rArr (r_1 cos theta_1, r_1 sin theta_1)`
`(x_2,y_2) rArr (r_2 cos theta_2, r_2 sin theta_2)`

so

`d = sqrt((x_1-x_2)^2+(y_1-y_2)^2) rArr sqrt((r_1 costheta_1-r_2 cos theta_2)^2+(r_1 sin theta_1-r_2 sin theta_2)^2)`

then

`d = sqrt(r_1^2+r_2^2-2r_1r_2(cos theta_1 cos theta_2+sin theta_1 sin theta_2)) = sqrt(r_1^2+r_2^2-2r_1r_2cos (theta_1 -theta_2))`