Chapter 14.1, Problem 57E

To determine

To show: The following relationships among the unit vectors in Cartesian and polar coordinates:

$u_r=\cos \theta \widehat{i}+\sin \theta \widehat{j}\text{, }i=u_r\cos \theta -u_{\theta }\sin \theta ,u_{\theta }=-\sin \theta \widehat{i}+\cos \theta \widehat{j}\text{ and }j=u_r\sin \theta +u_{\theta }\cos \theta$

are consistent with $\theta =0,\frac{\pi }{2},\pi \text{ and }\frac{3\pi }{2}$.