Chapter 11.3, Problem 48ES

Let u be a unit vector in the xy-plane of an xyz-coordinate system, and let v be a unit vector in the yz-plane. Let ${\theta }_1$ be the angle between u and i, let ${\theta }_2$ be the angle between v and k, and let $\theta$ be the angle between u and v.

(a) Show that $\cos \theta =\pm \sin {\theta }_1\sin {\theta }_2.$

(b) Find $\theta \text{if}\theta \text{is}\text{acute}\text{and}{\theta }_1={\theta }_2={45}^{\circ }.$

(c) Use a CAS to find, to the nearest degree, the maximum and minimum values of $\theta$ if $\theta$ is acute and ${\theta }_2=2{\theta }_1.$