Chapter 11 - Vectors and the Geometry of Space - Review Exercises - Page 811: 33

\[\begin{align} & \left( \mathbf{a} \right)-9\mathbf{i}+26\mathbf{j}-7\mathbf{k} \\ & \left( \mathbf{b} \right)9\mathbf{i}-26\mathbf{j}+7\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let }\mathbf{u}=4\mathbf{i}+3\mathbf{j}+6\mathbf{k},\text{ }\mathbf{v}=5\mathbf{i}+2\mathbf{j}+\mathbf{k} \\ & \\ & \left( \mathbf{a} \right)\text{Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 3 & 6 \\ 5 & 2 & 1 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 3 & 6 \\ 2 & 1 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 4 & 6 \\ 5 & 1 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 4 & 3 \\ 5 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=-9\mathbf{i}+26\mathbf{j}-7\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 2 & 1 \\ 4 & 3 & 6 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} 2 & 1 \\ 3 & 6 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 5 & 1 \\ 4 & 6 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 5 & 2 \\ 4 & 3 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=9\mathbf{i}-26\mathbf{j}+7\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 5 & 2 & 1 \\ 5 & 2 & 1 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} 2 & 1 \\ 2 & 1 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 5 & 1 \\ 5 & 1 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 5 & 2 \\ 5 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ \end{align}\]