Chapter 11 - Vectors and the Geometry of Space - 11.4 Exercises - Page 781: 9

\[\begin{align} & \left( \mathbf{a} \right)17\mathbf{i}-33\mathbf{j}-10\mathbf{k} \\ & \left( \mathbf{b} \right)-17\mathbf{i}+33\mathbf{j}+10\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let the vectors be: }\mathbf{u}=\left\langle 7,3,2 \right\rangle ,\text{ }\mathbf{v}=\left\langle 1,-1,5 \right\rangle \\ & \\ & \left( \mathbf{a} \right)\text{ Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 7 & 3 & 2 \\ 1 & -1 & 5 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 3 & 2 \\ -1 & 5 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 7 & 2 \\ 1 & 5 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 7 & 3 \\ 1 & -1 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=17\mathbf{i}-33\mathbf{j}-10\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{ Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -1 & 5 \\ 7 & 3 & 2 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} -1 & 5 \\ 3 & 2 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 1 & 5 \\ 7 & 2 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 1 & -1 \\ 7 & 3 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=-17\mathbf{i}+33\mathbf{j}+10\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{ Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -1 & 5 \\ 1 & -1 & 5 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ & \\ & \text{Summary} \\ & \left( \mathbf{a} \right)17\mathbf{i}-33\mathbf{j}-10\mathbf{k} \\ & \left( \mathbf{b} \right)-17\mathbf{i}+33\mathbf{j}+10\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]