Chapter 11 - Vectors and the Geometry of Space - 11.4 Exercises - Page 781: 10

\[\begin{align} & \left( \mathbf{a} \right)8\mathbf{i}-5\mathbf{j}+17\mathbf{k} \\ & \left( \mathbf{b} \right)-8\mathbf{i}+5\mathbf{j}-17\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let the vectors be: }\mathbf{u}=\left\langle 3,-2,-2 \right\rangle ,\text{ }\mathbf{v}=\left\langle 1,5,1 \right\rangle \\ & \\ & \left( \mathbf{a} \right)\text{ Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & -2 & -2 \\ 1 & 5 & 1 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} -2 & -2 \\ 5 & 1 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 3 & -2 \\ 1 & 1 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 3 & -2 \\ 1 & 5 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=8\mathbf{i}-5\mathbf{j}+17\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{ Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 5 & 1 \\ 3 & -2 & -2 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} 5 & 1 \\ -2 & -2 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 1 & 1 \\ 3 & -2 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 1 & 5 \\ 3 & -2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=-8\mathbf{i}+5\mathbf{j}-17\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{ Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 5 & 1 \\ 1 & 5 & 1 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ & \\ & \text{Summary} \\ & \left( \mathbf{a} \right)8\mathbf{i}-5\mathbf{j}+17\mathbf{k} \\ & \left( \mathbf{b} \right)-8\mathbf{i}+5\mathbf{j}-17\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]