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 - 5.2 Inner Product Spaces - 5.2 Exercises - Page 245: 21


(a) $\langle u, v \rangle =u_1v_1+u_1v_2+u_3v_3=0-18-16=-34$ (b) $\| u \| = \sqrt{97}$ (c) $\| v \| = \sqrt{101}$ (d) $d(u,v) =\sqrt{266}$

Work Step by Step

$u=(0,9,4), \quad v=(9,-2,-4)$ (a) $\langle u, v \rangle =u_1v_1+u_1v_2+u_3v_3=0-18-16=-34$ (b) $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2+u_3^2}=\sqrt{0+81+16}=\sqrt{97}$ (c) $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2+v_3^2}=\sqrt{81+4+16}=\sqrt{101}$ (d) $d(u,v)=\| u-v \|=\| (-9,11,8 \|=\sqrt{81 +121+64}=\sqrt{266}$
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