Textbook Problem

Distance Formula

(a) Verify that the Distance Formula for the distance between

the two points $(r_1,{\theta }_1)$ and $(r_2,{\theta }_2)$ in polar coordinates is $d=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos ({\theta }_1-{\theta }_2)}$.

(b) Describe the positions of the points relative to each other for ${\theta }_1={\theta }_2$, Simplify the Distance Formula for this case. Is the simplification what you expected? Explain.

(c) Simplify the Distance Formula for ${\theta }_1-{\theta }_2={90}^{\circ }$. Is the simplification what you expected? Explain.

(d) Choose two points on the polar coordinate system and find the distance between them. Then choose differentpolar representations of the same two points and apply the Distance Formula again. Discuss the result.