As has already been mentioned in the previous section, the vector product \(\vec v\) of three vectors is \(\vec a,\;\;\vec b,\;\;\vec c\) defined as
\[\vec v = \vec a \times (\vec b \times \vec c)\]
The most important property that the vector triple product satisfies is this:
\[\boxed{\begin{align}&\vec v = \vec a \times (\vec b \times \vec c) = (\vec a \cdot \vec c)\;\vec b - (\vec a \cdot \vec b)\vec c \hfill \\\\& \,\,\,\,\,\,\,\;\,{\text{I}}\,\,\,\,\,\,\,{\text{II}}\,\,\,\,{\text{III}}\,\,\,\,\,\,\,{\text{(I}} \cdot {\text{III)}}\,\,\,{\text{II -- }}\,{\text{(I}} \cdot {\text{II)III}}\hfill\\\end{align}}\qquad\qquad\qquad....\left( 1 \right)\]
Let us see how this property comes about. Since \(\vec v\) is perpendicular to \((\vec b \times \vec c)\) (and \(\vec a\) too), \(\vec v\) must lie in the plane containing \(\vec b\)and \(\vec c\) (convince yourself about this). We may \(\vec b\,\,{\text{and}}\,\,\vec c\) assume and to be non-collinear (since if they are collinear, (1) automatically holds because both sides will be zero). Thus, \(\vec b\,\,{\text{and}}\,\,\vec c\) form a basis of the plane in which they lie.
\[\vec v = \lambda \vec b + \mu \vec c\,\,for{\text{ }}some{\text{ }}\lambda ,\;\mu \in \mathbb{R}\qquad\qquad\qquad...\left( 2 \right)\]
Now, since \(\vec v\) is perpendicular to \(\vec a\) too, we have
\[\begin{align}&\quad\qquad\vec v \cdot \vec a = 0 \hfill \\\\& \Rightarrow \quad (\lambda \vec b + \mu \vec c) \cdot \vec a = 0 \hfill \\\\& \Rightarrow \quad \lambda (\vec a \cdot \vec b) + \mu (\vec a \cdot \vec c) = 0 \hfill \\\\& \Rightarrow\quad \frac{\lambda }{{(\vec a \cdot \vec c)}} = \frac{\mu }{{ - (\vec a \cdot \vec b)}} = l({\text{say}}) \hfill \\\\&\Rightarrow \quad \lambda = l(\vec a \cdot \vec c),\;\;\mu = - l(\vec a \cdot \vec b) \hfill \\ \end{align} \]
Using the values of \(\lambda \,and\,\mu \) in (2), we have
\[\vec v = l\{ (\vec a \cdot \vec c)\vec b - (\vec a \cdot \vec b)\vec c\} \qquad\qquad\qquad ...\left( 3 \right)\]
The only task that now remains is to find out the value of l. This can be done by taking particular values of \(\vec a,\,\,\vec b\,\,{\text{and}}\,\vec c\) ((3) must hold for all arbitrary vectors \(\vec a,\vec b,\vec c\) ).
Let \(\vec a = \hat i,\;\;\vec b = \hat i + \hat j\;\;{\text{and}}\;\;\vec c = \hat k\)
Thus,
\[\begin{align}&\qquad\vec v = \vec a \times (\vec b \times \vec c) \hfill \\\\&\;\qquad= \hat i \times \left\{ {(\hat i + \hat j) \times \hat k} \right\} \hfill \\\\&\;\qquad = \hat i \times \left\{ { - \hat j + \hat i} \right\} \hfill \\\\&\;\qquad= - \hat k \hfill \\\\&and\quad l\left\{ {(\vec a \cdot \vec c)\vec b - (\vec a \cdot \vec b)\vec c} \right\} = l\left\{ {0 - \hat k} \right\} \hfill \\\\&\;\qquad = - l\hat k \hfill \\ \end{align} \]
This gives l = 1. Thus, we see that the relation in (1) holds.
It should be obvious that the vector triple product is a non-associative operation. For example, \(\vec a \times (\vec b \times \vec c)\) lies in the plane of \(\vec b\) and \(\vec c\) while \((\vec a \times \vec b) \times \vec c = ( - \vec c) \times (\vec a \times \vec b)\) lies in the plane of \(\vec a\) and \(\vec b\) .
In general, therefore,
\[\vec a \times (\vec b \times \vec c) \ne (\vec a \times \vec b) \times \vec c\]
