2022 ACJC Promo Q5

The line \(L\) has vector equation

\(L:\mathbf{r}=\left( \begin{matrix} 6 \\ 0 \\ 1 \\ \end{matrix} \right)+\lambda \,\,\left( \begin{matrix} -2 \\ 1 \\ -1 \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).

1. Find the position vector of the point \(N\), the foot of perpendicular from the point \(A\) with coordinates \(\left( 3,\,6,\,1 \right)\) to the line \(L\).[3]
2. Find the position vector of the point \(A'\), the reflection of point \(A\) in the line \(L\).[2]
3. Find the coordinates of the points on the line \(L\) that are \(3\sqrt{3}\) units away from point \(A\).[2]