Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 5 - Inner Product Spaces - Review Exercises - Page 284: 17


$\theta = \frac{\pi}{12} +2n\pi$ and $n$ is an integer.

Work Step by Step

Let $u=\left(\cos \frac{3\pi}{4},\sin \frac{3\pi}{4}\right),\quad v=\left(\cos \frac{2\pi}{3},\sin \frac{2\pi}{3}\right) \quad \langle u, v\rangle=u\cdot v $. The angle $\theta$ between $u$ and $v$ is given by the formula $$\cos \theta=\frac{\langle u, v\rangle}{\|u\| \cdot\|v\|}=\frac{\cos \frac{3\pi}{4}\cos \frac{2\pi}{3}+\sin \frac{3\pi}{4}\sin \frac{2\pi}{3}}{\sqrt{ \cos^2 \frac{3\pi}{4}+\sin^2 \frac{3\pi}{4}}\sqrt{ \cos^2 \frac{2\pi}{3}+\sin^2 \frac{2\pi}{3}}}.$$ Using the properties of the trignometric functions, we get $$\cos \theta=\cos\left( \frac{3\pi}{4}- \frac{2\pi}{3}\right)=\cos \frac{\pi}{12} $$ that is $\theta = \frac{\pi}{12} +2n\pi$ and $n$ is an integer.
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