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: 5


(a) $\| u \| = \sqrt{6}$ (b) $\| v \| = =\sqrt{3}$ (c) $\langle u, v \rangle = -1$ (d) $d(u,v)= \sqrt{11}.$

Work Step by Step

Let $u=(1,-2,0,1), \quad v=(1,1,-1,0)$, then we have (a) $\| u \| =\sqrt{\langle u, u\rangle}=\sqrt{u_1^2+u_2^2+u_3^2}=\sqrt{6}$ (b) $\| v \| =\sqrt{\langle v, v\rangle}=\sqrt{v_1^2+v_2^2+v_3^2}=\sqrt{3}$ (c) $\langle u, v \rangle =u_1v_1+u_2v_2+u_3v_3= -1$ (d) $d(u,v)=\| u-v \|=\| (0,-3,1,1) \|=\sqrt{11}.$
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