Chapter 11 - Vectors and the Geometry of Space - 11.4 Exercises - Page 781: 7

\[\begin{align} & \left( \mathbf{a} \right)20\mathbf{i}+10\mathbf{j}-16\mathbf{k} \\ & \left( \mathbf{b} \right)-20\mathbf{i}-10\mathbf{j}+16\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]

\[\begin{align} & \text{Let the vectors: }\mathbf{u}=-2\mathbf{i}+4\mathbf{j},\text{ }\mathbf{v}=\text{3}\mathbf{i}\text{+2}\mathbf{j}\text{+5}\mathbf{k} \\ & \\ & \left( \mathbf{a} \right)\text{ Find }\mathbf{u}\times \mathbf{v} \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -2 & 4 & 0 \\ 3 & 2 & 5 \\ \end{matrix} \right| \\ & \mathbf{u}\times \mathbf{v}=\left| \begin{matrix} 4 & 0 \\ 2 & 5 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} -2 & 0 \\ 3 & 5 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} -2 & 4 \\ 3 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{u}\times \mathbf{v}=20\mathbf{i}+10\mathbf{j}-16\mathbf{k} \\ & \\ & \left( \mathbf{b} \right)\text{ Find }\mathbf{v}\times \mathbf{u} \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 2 & 5 \\ -2 & 4 & 0 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{u}=\left| \begin{matrix} 2 & 5 \\ 4 & 0 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 3 & 5 \\ -2 & 0 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 3 & 2 \\ -2 & 4 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{u}=-20\mathbf{i}-10\mathbf{j}+16\mathbf{k} \\ & \\ & \left( \mathbf{c} \right)\text{ Find }\mathbf{v}\times \mathbf{v} \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 2 & 5 \\ 3 & 2 & 5 \\ \end{matrix} \right| \\ & \mathbf{v}\times \mathbf{v}=\left| \begin{matrix} 2 & 5 \\ 2 & 5 \\ \end{matrix} \right|\mathbf{i}-\left| \begin{matrix} 3 & 5 \\ 3 & 5 \\ \end{matrix} \right|\mathbf{j}+\left| \begin{matrix} 3 & 2 \\ 3 & 2 \\ \end{matrix} \right|\mathbf{k} \\ & \mathbf{v}\times \mathbf{v}=\mathbf{0} \\ & \\ & \text{Summary} \\ & \left( \mathbf{a} \right)20\mathbf{i}+10\mathbf{j}-16\mathbf{k} \\ & \left( \mathbf{b} \right)-20\mathbf{i}-10\mathbf{j}+16\mathbf{k} \\ & \left( \mathbf{c} \right)\mathbf{0} \\ \end{align}\]