If$b = i – j + 3k, c = j + 2k$ and $a$ is a unit vector, then the maximum value of the scalar triple product $[a b c]$ is

$\begin{array}{r}\mathbf{b}\times \mathbf{c}=\left|\begin{array}{ccc}\mathit{i}& \mathit{j}& \mathit{k}\\ 1& -1& 3\\ 0& 1& 2\end{array}\right|\\ =-5\mathit{i}-2\mathit{j}+\mathit{k}\end{array}$

Now, $\begin{array}{l}\left[\mathbf{abc}\right]=\mathbf{a}\cdot (\mathbf{b}\times \mathbf{c})\\ =\left|\mathbf{a}\right||\mathbf{b}\times \mathbf{c}|\cos \theta \end{array}$

where $\theta =$ angle between $a$ and $b \times c$

Max$[a b c] = \sqrt{30}$ when $\cos \theta = 1$