2023 MI PRU1 P1 Q7

Relative to the origin \(O\), the point \(A\) has position vector \(3\mathbf{i}+7\mathbf{j}+\mathbf{k}\). The plane \(\pi \) contains \(A\) and is parallel to the vector \(-2\mathbf{i}+2\mathbf{j}+\mathbf{k}\) and the line \(\mathbf{r}=\alpha \left( -\mathbf{i}+2\mathbf{j}+3\mathbf{k} \right)\), where \(\alpha \in \mathbb{R}\).

1. Show that the cartesian equation of plane \(\pi \) can be expressed as \(4x+5y-2z=45\).[3]

The point \(B\) has coordinates \(\left( 2,\ 0,\ 4 \right)\).

2. Find the position vector of the point \(F\), the foot of the perpendicular from \(B\) to the plane \(\pi \).[3]
3. Hence find the position vector of the point \({B}'\), the reflection of \(B\) in the plane \(\pi \).[2]
4. Find angle \(BA{B}'\).[3]