2017 CJC J1 MYE Q8

The plane \({{p}_{1}}\) contains points \(A\) and \(B\), with position vectors \(\left( \begin{matrix}2 \\0 \\1 \end{matrix} \right)\) and \(\left( \begin{matrix}8 \\-6 \\3 \end{matrix} \right)\) respectively, and is parallel to the line with direction vector \(\left( \begin{matrix}7 \\-5 \\3 \end{matrix} \right)\).

1. Find a cartesian equation of the plane \({{p}_{1}}\).[3]

The plane \({{p}_{2}}\) has equation \(\mathbf{r}\cdot (\mathbf{i}-2\mathbf{j}+\mathbf{k})=3\). The planes \({{p}_{1}}\) and \({{p}_{2}}\) meet in the line \(l\).

2. Find a vector equation of the line \(l\).[2]
3. Find the acute angle between the planes \({{p}_{1}}\) and \({{p}_{2}}\).[2]

It is given that the point \(R\) with position vector \(-17\mathbf{i}-5\mathbf{j}+10\mathbf{k}\) lies on \({{p}_{2}}\).

4. Find the foot of perpendicular from the point \(R\) to the plane \({{p}_{1}}\).[3]
5. Hence, find a vector equation of the plane \({{p}_{3}}\) which is a reflection of the plane \({{p}_{2}}\) in the plane \({{p}_{1}}\) in the form \(\mathbf{r}\cdot \mathbf{n}=d\), where \(d\) is a constant to be determined.[4]