If the volume of the parallelopiped formed by three non-coplanar vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) is 4 cu units, then \([\mathbf{a} \times \mathbf{b} \mathbf{b} \times \mathbf{c} \mathbf{c} \times \mathbf{a}]\) is equal to

In scalar triple product, the position of dot and cross can be changed provided the cyclic order is maintained i.e.,
\[
[\mathbf{a} \mathbf{b} \mathbf{c}]=(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}=\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c})
\]
Put \(\mathbf{c} \times \mathbf{a}=\mathbf{n}\)
\(\therefore[\mathbf{a} \times \mathbf{b} \mathbf{b} \times \mathbf{c} \mathbf{c} \times \mathbf{a}]=(\mathbf{a} \times \mathbf{b}) \cdot\{(\mathbf{b} \times \mathbf{c}) \times \mathbf{n}\}\)
\(=(\mathbf{a} \times \mathbf{b}) \cdot\{(\mathbf{n} \cdot \mathbf{b}) \mathbf{c}-(\mathbf{n} \cdot \mathbf{c}) \mathbf{b}\}\)
\(=(\mathbf{a} \times \mathbf{b}) \cdot[\{(\mathbf{c} \times \mathbf{a}) \cdot \mathbf{b}\} \mathbf{c}-\{(\mathbf{c} \times \mathbf{a}) \cdot \mathbf{c}\} \mathbf{b}]\)
\(=(\mathbf{a} \times \mathbf{b}) \cdot\{[\mathbf{c} \mathbf{a} \mathbf{b}] \mathbf{c}-[\mathbf{c} \mathbf{a} \mathbf{c}] \mathbf{b}\}\)
\(=[(\mathbf{a} \times \mathbf{b}) \cdot \mathbf{c}][\mathbf{c} \mathbf{a} \mathbf{b}]-0\)
\(=[\mathbf{a b c}]\) [abc]
\(=[\mathbf{a b c}]^{2}=(4)^{2}=16\)
Alternate Method
By properties of Scalar triple product,
\([\mathbf{a} \times \mathbf{b} \mathbf{b} \times \mathbf{c} \mathbf{c} \times \mathbf{a}]=\left[\begin{array}{lll}\mathbf{a} & \mathbf{b} & \mathbf{c}^{2}\end{array}\right.\)
\[
=(4)^{2}=16
\]