2025 ASRJC Promo Q10

The plane \(p_1\) has equation \(x+y+z=3\) while the line \(l_1\) passes through the origin and is parallel to \(-2\mathbf{i}+3\mathbf{j}-2\mathbf{k}\). It is given that the point \(A\) has position vector \(\mathbf{i}\).

1. Find the acute angle, in degrees, between \(l_1\) and \(p_1\).[2]
2. Find the exact coordinates of the foot of perpendicular from the point \(A\) to \(p_1\).[4]
3. Point \(B\) is a point on the line \(l_1\). It is given that the shortest distance from point \(B\) to the plane \(p_1\) is \(\sqrt{27}\) units, find the possible coordinates of \(B\).[3]

The plane \(p_2\) has equation \(3x+4y+\alpha z=7+\alpha\), \(\alpha \in \mathbb{R}\).

4. It is given that the point \(C\) with coordinates \((1,1,1)\) lies on \(p_1\). Show that point \(C\) also lies on \(p_2\). Hence find a vector equation of the line \(l_2\), the line of intersection between \(p_1\) and \(p_2\), in terms of \(\alpha\).[3]