Chapter 11 - Vectors and the Geometry of Space - Review Exercises - Page 811: 35

\[\begin{align} & \left( \mathbf{a} \right)-8\mathbf{i}-10\mathbf{j}+6\mathbf{k} \\ & \left( \mathbf{b} \right)8\mathbf{i}+10\mathbf{j}-6\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let }\mathbf{u}=\left\langle 2,-4,-4 \right\rangle ,\text{ }\mathbf{v}=\left\langle 1,1,3 \right\rangle \\ & \\ & \left( \mathbf{a} \right)\text{Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -4 & -4 \\ 1 & 1 & 3 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} -4 & -4 \\ 1 & 3 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 2 & -4 \\ 1 & 3 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 2 & -4 \\ 1 & 1 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=-8\mathbf{i}-10\mathbf{j}+6\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 3 \\ 2 & -4 & -4 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} 1 & 3 \\ -4 & -4 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 1 & 3 \\ 2 & -4 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 1 & 1 \\ 2 & -4 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=8\mathbf{i}+10\mathbf{j}-6\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 1 & 3 \\ 1 & 1 & 3 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ \end{align}\]