2023 NJC P2 Q1

Relative to the origin, the fixed point \(A\) has position vector \(\mathbf{a}\). The line \(l\) has equation \(\mathbf{r}=s\mathbf{d}\), where \(s\) is a parameter and \(\mathbf{d}\) is a unit vector not parallel to \(\mathbf{a}\).

1. Interpret geometrically the vector equation \(\mathbf{r}=\mathbf{a}+t\,\mathbf{d}\), \(t\in \mathbb{R}\).[2]
2. Find, in terms of \(\mathbf{a}\) and \(\mathbf{d}\), the position vector of the foot of perpendicular from \(A\) to \(l\).[1]
3. A variable point \(P\) with position vector \(\mathbf{p}\) lies on the plane containing \(A\) and \(l\). Describe the locus of \(P\) if \(\left| \mathbf{p}\times \mathbf{d} \right|=\left| \mathbf{a}\times \mathbf{d} \right|\).[2]