Question

This problem concerns the connections between the polar plane and the θr-plane.

(a) Which points in the θr-plane correspond to the pole in the polar plane, and why?

(b) Which points in the θr-plane correspond to the horizontal axis in the polar plane, and why?

(c) Which points in the θr-plane correspond to the vertical axis in the polar plane, and why?

Short answer

a)$\left(\theta ,0\right)$corresponds to the pole.

b) $\left(0,r\right),\left(\mathrm{\pi },-\mathrm{r}\right)$corresponds to the points on the horizontal axis.

c)$\left(\frac{\mathrm{\pi }}{2},r\right),\left(\frac{3\mathrm{\pi }}{2},-r\right)$corresponds to points on the vertical axis.

Step-by-step solution

Step 1. Given information

Two different planes: polar plane and$\theta r$plane.

Part (a) Step 1.  Find the points in the θr-plane correspond to the pole in the polar plane 

The pole in polar form is the point where $r=0$

So the point in$\theta r$plane islocalid="1653109322792" $\left(0,0\right)$

Part (b) Step 1. Find points in the θr-plane correspond to the horizontal axis in the polar plane:

In the polar plain the points on horizontal axis is $\left(r,0\right),\left(-r,\mathrm{\pi }\right)$

In $\theta r$plane these points correspond to point$\left(0,r\right),\left(\mathrm{\pi },-\mathrm{r}\right)$

Part (c) Step 1. Find points in the θr-plane correspond to the vertical axis in the polar plane.

In polar plane the points on vertical lines are $\left(r,\frac{\mathrm{\pi }}{2}\right),\left(-r,\frac{3\mathrm{\pi }}{2}\right)$

In $\theta r$plane that points corresponds to the point$\left(\frac{\mathrm{\pi }}{2},r\right),\left(\frac{3\mathrm{\pi }}{2},-r\right)$