Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 12 - Section 12.4 - The Cross Product - 12.4 Exercises - Page 821: 26


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

Work Step by Step

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