2025 ACJC Promo Q10

The plane \(p_1\) has equation \(x-y+z=0\) and the plane \(p_2\) has equation \(x+y+z=2\).

1. Find the acute angle between planes \(p_1\) and \(p_2\).[2]
2. Find the vector equation of the line of intersection of planes \(p_1\) and \(p_2\).[1]

With reference to the origin \(O\), the points \(A\) and \(B\) are such that \(\overrightarrow{OA}=-2\mathbf{i}+\mathbf{j}-\mathbf{k}\) and \(\overrightarrow{OB}=\mathbf{i}+\mathbf{j}\). The line \(l\) has equation \(\mathbf{r}=-2\mathbf{i}+\mathbf{j}-\mathbf{k}+\mu(\mathbf{i}-\mathbf{k})\), where \(\mu\) is a parameter.

3. Find the shortest distance between line \(l\) and the line with cartesian equation \(x-3=-z+2\), \(y=1\).[2]
4. The point \(C\) lies on \(l\) such that angle \(ABC=90^\circ\). Find the position vector of point \(C\).[3]
5. Find the position vector of the foot of perpendicular from point \(A\) to \(p_2\).[3]
6. A plane \(p_3\) is equidistant to both point \(A\) and plane \(p_2\). Find the cartesian equation of plane \(p_3\).[2]