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 861: 25


$a\times (b+c)=a \times b+a \times c$

Work Step by Step

Let $a= a_1i+a_2j+a_3k$; $b=b_1i+b_2j+b_3k$ and $c=c_1i+c_2j+c_3k$ $a\times (b+c)=\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1+c_1&b_2+c_2&b_3+c_3\end{vmatrix}$ Using property of determinants,we can write $= \begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}+\begin{vmatrix} i&j&k \\ a_1&a_2&a_3\\c_1&c_2&c_3\end{vmatrix}$ $=a \times b+a \times c$ Hence, $a\times (b+c)=a \times b+a \times c$
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