## Precalculus (10th Edition)

$v\times w=i+5j-7k.$ $w\times v=-i-5j+7k.$ $w\times w=v\times v=0.$
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 $D=v\times w=i((3\cdot(-1)-2\cdot(-2))-j((-1)\cdot(-1)-2\cdot3)+k((-1)\cdot(-2)-3\cdot3)=i+5j-7k.$ We know that $w\times v=-v\times w=-(i+5j-7k)=-i-5j+7k.$ We also know that $w\times w=v\times v=0.$