Chapter 11 - Vectors and the Geometry of Space - Review Exercises - Page 811: 34

\[\begin{align} & \left( \mathbf{a} \right)-19\mathbf{i}-26\mathbf{j}-8\mathbf{k} \\ & \left( \mathbf{b} \right)19\mathbf{i}+26\mathbf{j}+8\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let }\mathbf{u}=6\mathbf{i}-5\mathbf{j}+2\mathbf{k},\text{ }\mathbf{v}=-4\mathbf{i}+2\mathbf{j}+3\mathbf{k} \\ & \\ & \left( \mathbf{a} \right)\text{Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 6 & -5 & 2 \\ -4 & 2 & 3 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} -5 & 2 \\ 2 & 3 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 6 & 2 \\ -4 & 3 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 6 & -5 \\ -4 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=-19\mathbf{i}-26\mathbf{j}-8\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & 2 & 3 \\ 6 & -5 & 2 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} 2 & 3 \\ -5 & 2 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} -4 & 3 \\ 6 & 2 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} -4 & 2 \\ 6 & -5 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=19\mathbf{i}+26\mathbf{j}+8\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -4 & 2 & 3 \\ -4 & 2 & 3 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ \end{align}\]