Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 13 - Vector Geometry - 13.4 The Cross Product - Exercises - Page 677: 23


$$ v\times i =cj-bk $$ $$ v\times j= -ci+ak.$$ $$ v\times k= bi-aj.$$

Work Step by Step

First, we can write $$ v=ai+bj+ck.$$ Now, we have $$ v\times i=(ai+bj+ck)\times i\\ =a i\times i+b j\times i+c k\times i\\ =-bk+cj=cj-bk $$ $$ v\times j=(ai+bj+ck)\times j\\ =a i\times j+b j\times j+c k\times j\\ =ak-ci=-ci+ak.$$ $$ v\times k=(ai+bj+ck)\times k\\ =a i\times k+b j\times k+c k\times k\\ =-aj+bi=bi-aj.$$
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