Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 9 - Polar Coordinates; Vectors - 9.7 The Cross Product - 9.7 Assess Your Understanding - Page 631: 19

Answer

$v\times w=i((-1)\cdot(-1)-2\cdot1)-j(2\cdot(-1)-2\cdot0)+k(2\cdot1-(-1)\cdot0)=-i+2j+2k.$ We know that $w\times v=-v\times w=-(-i+2j+2k)=i-2j-2k.$ We also know that $w\times w=v\times v=0.$

Work Step by Step

We know that for a matrix \[ \left[\begin{array}{rrr} a & b & c \\ d &e & f \\ g &h & i \\ \end{array} \right] \] the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$ We know that if we have two vectors $v=ai+bj+ck$ and $w=di+ej+fk$, then $v\times w$ can be obtained by the determinant of: \[ \left[\begin{array}{rrr} i & j & k \\ a &b & c \\ d &e & f \\ \end{array} \right] \] Hence here $D=v\times w=i((-1)\cdot(-1)-2\cdot1)-j(2\cdot(-1)-2\cdot0)+k(2\cdot1-(-1)\cdot0)=-i+2j+2k.$ We know that $w\times v=-v\times w=-(-i+2j+2k)=i-2j-2k.$ We also know that $w\times w=v\times v=0.$
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