2025 MI P1 Q9

The coordinates of point \(A\) and \(B\) are \((1, 2, 3)\) and \((17, 10, 19)\) respectively. The line \(l_1\) passes through both points \(A\) and \(B\) and is parallel to the vector \(\begin{pmatrix} k \\ 1 \\ 2 \end{pmatrix}\), where \(k\) is a constant.

1. Show that \(k = 2\).[2]

The line \(l_2\) has equation \(\mathbf{r} = \begin{pmatrix} 3 \\ 4 \\ 6 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}\) where \(\mu\) is a parameter.

2. Determine whether the lines \(l_1\) and \(l_2\) intersect. If they do, find the position vector of their point of intersection; otherwise, explain why they do not intersect.[4]

The plane \(p\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = 16\).

3. Let \(\alpha\) be the acute angle between \(l_1\) and \(p\). Find the exact value of \(\sin \alpha\) and \(\cos \alpha\).[3]
4. Find the coordinates of point \(F\), the foot of the perpendicular of the point \(B\) to the plane \(p\).[3]