2023 CJC Promo Q3

Plane \({{\pi }_{1}}\) contains point \(A\left( 7,5,1 \right)\) and line \(l\) with equation \(\mathbf{r}=\left( \begin{matrix} -5 \\ 5 \\ 4 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 4 \\ 2 \\ 1 \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).

1. Find an equation of \({{\pi }_{1}}\) in scalar product form.[3]

Plane \({{\pi }_{2}}\) has equation \(x+3y-3z=8\).

2. Find the acute angle between \({{\pi }_{1}}\) and \({{\pi }_{2}}\).[2]
3. Find the perpendicular distance from \(A\) to \({{\pi }_{2}}\).[2]