2025 JPJC P2 Q2

With reference to the origin \(O\), the points \(A\), \(P\) and \(Q\) have position vectors \(\mathbf{a}\), \(\mathbf{p}\) and \(\mathbf{q}\) respectively. A straight line \(l\) through the point \(A\) is parallel to a unit vector \(\mathbf{e}\).

1. The point \(Q\) lies on \(l\). Show that \((\mathbf{q} - \mathbf{a}) \times \mathbf{e} = \mathbf{0}\).[2]
2. Given that \(P\) is not on \(l\), give the geometrical meaning of \(|(\mathbf{p} - \mathbf{a}) \times \mathbf{e}|\).[1]

It is also given that \(\mathbf{p} = 3\mathbf{q}\) and \(AQ\) is \(2\) units.

3. Using the results in (a) and (b), or otherwise, show that the area of triangle \(APQ\) can be written as \(k|\mathbf{q} \times \mathbf{e}|\), where \(k\) is a constant to be determined.[4]
4. Hence, or otherwise, find the acute angle between \(l\) and \(PQ\), given further that the area of triangle \(\Delta APQ\) is \(3\) units\({}^2\) and \(|\mathbf{q}| = 2\).[3]