\(\vec{a}=\overrightarrow{3 i}+\overrightarrow{4 j}-\overrightarrow{5 k}, \vec{b}=\overrightarrow{5 i}-\overrightarrow{3j}+\overrightarrow{6 k}\)
\(\vec{a}+\vec{b}=\overrightarrow{10 i}+\vec{j}+\vec{k}\)
\(\vec{a}-\vec{b}=\overrightarrow{4 i}+ \overrightarrow{7j}+\vec{k}\)
\(\therefore \quad(\vec{a}+\vec{b}) \times(\vec{a}-\vec{b})=\left|\begin{array}{ccc}\vec{i} & \vec{j} & \vec{k} \\ 10 & 1 & 1 \\ -4 & 7 & 11\end{array}\right|\)
\(=\vec{a}(-11-7)-\vec{i}(-110+4) \times \vec{k}(70+4)\)
\(=\overrightarrow{17 i}+\overrightarrow{106 j}+\overrightarrow{74 k}\)
