2026 NYJC P1 Q11

The lines \(l_1\) and \(l_2\) have equations \[\mathbf{r}=\begin{pmatrix}2\\1\\-1\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\3\end{pmatrix}\qquad\text{and}\qquad\mathbf{r}=\begin{pmatrix}3\\4\\p\end{pmatrix}+\mu\begin{pmatrix}2\\1\\q\end{pmatrix}\] respectively, where \(p\) and \(q\) are constants, and \(\lambda\) and \(\mu\) are parameters.

1. Suppose that \(l_1\) intersects \(l_2\).
   1. Find the coordinates of the point of intersection and state a linear relationship between \(p\) and \(q\).[3]
   2. Find the acute angle between \(l_1\) and \(l_2\) when \(q=2\).[2]
2. By using vector product, show that a vector perpendicular to both \(l_1\) and \(l_2\) is \(\begin{pmatrix}2q+3\\q-6\\m\end{pmatrix}\), where \(m\) is a constant to be determined.[2]
3. The plane \(\pi\) contains \(l_2\) and is parallel to \(l_1\). Find the shortest distance between \(l_1\) and \(\pi\) in terms of \(p\) and \(q\).[4]