2025 SAJC Promo Q10

The planes \(\pi_1\) and \(\pi_2\) have equations

\({2x + 2y + z = 2,}\)
\({5x - 2y - z = 5,}\)

respectively.

1. Find the acute angle between \(\pi_1\) and \(\pi_2\).[2]
2. Verify that the point \(P\) with position vector \(\mathbf{i}+\mathbf{j}-2\mathbf{k}\) lies in \(\pi_1\) and \(\pi_2\). Hence, find a vector equation of the line of intersection of \(\pi_1\) and \(\pi_2\).[3]
3. The point \(Q\) has position vector \(\alpha\mathbf{i}+2\alpha\mathbf{k}\), where \(\alpha \in \mathbb{R}\). Show that the perpendicular distance from \(Q\) to \(\pi_1\) is \(\dfrac{2|2\alpha-1|}{3}\).[3]

4. Hence or otherwise, find the cartesian equations of the two planes that are parallel to \(\pi_1\) and at a distance of \(6\) units from \(\pi_1\).

   [4]