CJC Vectors Tutorial Q2

Determine for each of the following pairs of line and plane, whether the line \(l\) intersects the plane \(\prod \) at a unique point, lies on the plane or do not intersect the plane. If the line intersects the plane at a unique point, find the point of intersection and the acute angle between the \(l\) and \(\prod \).

1. \(\prod :\mathbf{r}\cdot \left( \begin{matrix} 2 \\ 1 \\ -3 \\ \end{matrix} \right)=5\) and \(l:\mathbf{r}=\left( \begin{matrix} 3 \\ -1 \\ 1 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} -2 \\ 1 \\ -3 \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).
2. \(\prod :\mathbf{r}\cdot \left( \begin{matrix} 2 \\ 1 \\ -3 \\ \end{matrix} \right)=5\) and \(l:\mathbf{r}=\left( \begin{matrix} 2 \\ 1 \\ 0 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 1 \\ 1 \\ \end{matrix} \right)\), \(\lambda \in \mathbb{R}\).