Chapter 11 - Vectors and the Geometry of Space - Review Exercises - Page 811: 36

\[\begin{align} & \left( \mathbf{a} \right)11\mathbf{i}+\mathbf{j}-2\mathbf{k} \\ & \left( \mathbf{b} \right)-11\mathbf{i}-\mathbf{j}+2\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let }\mathbf{u}=\left\langle 0,2,1 \right\rangle ,\text{ }\mathbf{v}=\left\langle 1,-3,4 \right\rangle \\ & \\ & \left( \mathbf{a} \right)\text{Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 0 & 2 & 1 \\ 1 & -3 & 4 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 2 & 1 \\ -3 & 4 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 0 & 1 \\ 1 & 4 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 0 & 2 \\ 1 & -3 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=11\mathbf{i}+\mathbf{j}-2\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -3 & 4 \\ 0 & 2 & 1 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} -3 & 4 \\ 2 & 1 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 1 & 4 \\ 0 & 1 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 1 & -3 \\ 0 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=-11\mathbf{i}-\mathbf{j}+2\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -3 & 4 \\ 1 & -3 & 4 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ \end{align}\]