Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 12 - Vectors and the Geometry of Space - 12.4 The Cross Product - 12.4 Exercises - Page 862: 49


$(a-b) \times (a+b)= 2(a \times b)$

Work Step by Step

$(a-b) \times (a+b)=a \times (a+b)-b \times (a+b)$ $(a-b) \times (a+b)=a \times (a+b)-b \times (a+b)= a \times a+ a \times b - b \times a- b \times b$ As we know the cross product of a vector with itself is $0$. $(a-b) \times (a+b)=a \times (a+b)-b \times (a+b)= a \times a+ a \times b - b \times a- b \times b= 0+ a \times b - b \times a- 0$ Note: $ -b \times a =a \times b$ $=a \times b + a \times b$ $= 2(a \times b)$ Hence, $(a-b) \times (a+b)= 2(a \times b)$
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