2021 DHS P2 Q5 Modified

The position vectors of the variable point \(R\) and the fixed points \(A\) and \(B\) are given by

\(\mathbf{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}, \mathbf{a} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} \text{ and}\hspace{0.5em} \mathbf{b} = \begin{pmatrix} -1 \\ 5 \\ 3 \end{pmatrix} \text{ respectively.}\)

It is given that \(\mathbf{r} \cdot (\mathbf{a} \times \mathbf{b}) = 0\).