## Precalculus (10th Edition)

$-4i-2j+2k.$
We know that a vector will be orthogonal to two vectors if and only if it is their cross-product (multiplied by any constant). 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 $D=u\times (j+k)=i((-3)\cdot1-1\cdot1)-j(2\cdot1-1\cdot0)+k(2\cdot1-(-3)\cdot0)=-4i-2j+2k.$