Subsection 1.1.7 Vector Triple Product
If \(\vec{a}\text{,}\) \(\vec{b}\text{,}\) and \(\vec{c}\) be three vectors, then their vector triple product is denoted by \(\vec{a}\times{(\vec{b}\times\vec{c})}\text{.}\) Let \(\vec{a} = a_{1}\hat{i}+a_{2}\hat{j}+ a_{3}\hat{k}, \) \(\vec{b}= b_{1}\hat{i}+b_{2}\hat{j}
+ b_{3}\hat{k},\) and \(\vec{c} = c_{1}\hat{i}+c_{2}\hat{j}+ c_{3}\hat{k},\) then
\begin{equation*}
\vec{a}\times{(\vec{b}\times\vec{c})} = (a_{1}\hat{i}+a_{2}\hat{j}+ a_{3}\hat{k})
\times{\begin{Vmatrix} \hat{i} & \hat{j} & \hat{k} \\ b_{1} & b_{2} & b_{3}\\c_{1} &
c_{2} & c_{3} \end{Vmatrix}}
\end{equation*}
\begin{equation*}
=(a_{1}\hat{i}+a_{2}\hat{j}+ a_{3}\hat{k}) \times[(b_{2}c_{3}-b_{3}c_{2})\hat{i} + (b_{1}c_{3}-b_{3}c_{1})\hat{j} + (b_{1}c_{2}-b_{2}c_{1})\hat{k}]
\end{equation*}
\begin{equation*}
=[a_{2}(b_{1}c_{2}-b_{2}c_{1}) - a_{3}(b_{3}c_{1}-b_{1}c_{3})]\hat{i} + [a_{3}(b_{2}c_{3}-b_{3}c_{2})-a_{1}(b_{1}c_{2} - b_{2}c_{1})]\hat{j}
\end{equation*}
\begin{equation*}
+ [a_{1}(b_{3}c_{1}-b_{1}c_{3}) - a_{2}(b_{2}c_{3}-b_{3}c_{2})]\hat{k}
\end{equation*}
\begin{equation*}
=\left(a_{1}c_{1} + a_{2}c_{2} + a_{3}c_{3}\right) \left(b_{1}\hat{i} + b_{2}\hat{j}
+ b_{3}\hat{k}\right)
\end{equation*}
\begin{equation*}
- \left(a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3}\right) \left(c_{1}\hat{i} + c_{2}\hat{j} + c_{3}\hat{k}\right)
\end{equation*}
\begin{equation*}
=\left( \vec{a}\cdot\vec{c}\right)\vec{b} - \left( \vec{a}\cdot\vec{b}\right)\vec{c}
\end{equation*}
