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 246: 69

Answer

(a) The orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} =\frac{4}{5}(2,1).$$ (b) The orthogonal projection of $v$ onto $u$ is given by $$\operatorname{proj}_{{u}} {v} =\frac{4}{5}( 1,2).$$

Work Step by Step

Let $u=(1,2)$, $v=(2,1)$, $\langle{u}, {v}\rangle=u\cdot v$. Then, we have $$\langle{u}, {v}\rangle=4, \quad \langle{u}, {u}\rangle=5, \quad \langle{v}, {v}\rangle=5.$$ (a) The orthogonal projection of $u$ onto $v$ is given by $$\operatorname{proj}_{{v}} {u} =\frac{\langle{u}, {v}\rangle}{\langle{v}, {v}\rangle} {v}=\frac{4}{5}(2,1).$$ (b) The orthogonal projection of $v$ onto $u$ is given by $$\operatorname{proj}_{{u}} {v} =\frac{\langle{u}, {v}\rangle}{\langle{u}, {u}\rangle} {u}=\frac{4}{5}( 1,2).$$ (c)
Small 1566056761
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.