Linear Algebra: A Modern Introduction

Published by Cengage Learning
ISBN 10: 1285463242
ISBN 13: 978-1-28546-324-7

Chapter 1 - Vectors - 1.2 Length and Angle: The Dot Product - Exercises 1.2 - Page 290: 29


$\Theta \;=\;14^\circ 20^{'}$

Work Step by Step

$u=[1\;,2\;,3\;,4]\;\;\;\;\;\;\;\;\;v=\;[5\;,6\;,7\;,8]\\\\$ $cos \;\Theta\;=\frac{u\;.\;v}{\left \| u \right \|\;.\;\left \| v \right \|} \\\\$ $u\;.v\;=\;(1)(5)\;+\;(2)(6)\;+\;(3)(7)\;+\;(4)(8)\;=\;70\\\\$ $\left \| u \right \|\;=\;\sqrt{(1)^2\;+(2)^2\;+(3)^2\;+(4)^2}\;=\;\sqrt{30}\\\\$ $\left \| v \right \|\;=\;\sqrt{(5)^2\;+(6)^2\;+(7)^2\;+(8)^2}\;=\;\sqrt{174}\\\\$ $cos \;\Theta\;=\frac{u\;.\;v}{\left \| u \right \|\;.\;\left \| v \right \|}\;=\;\frac{70}{(\sqrt{30})(\sqrt{174})}\\\\$ $cos \;\Theta\;=\frac{7\sqrt{145}}{87}\\\\$ $\Theta \;=\;cos^{-1}(\frac{7\sqrt{145}}{87})\;=\;14^\circ 20^{'}$
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