A line, l_1, and a plane p_1, have equations $\mathbf{r} = \begin{pmatrix}1\\-2\end{pmatrix} + \lambda \begin{pmatrix}3\\2\end{pmatrix}, \lambda \in \mathbb{R}$ and $\mathbf{r} \cdot \begin{pmatrix}-1\\0\\2\end{pmatrix} = 4$ respectively.

1. (i)

   Find the intersection point of l_1 and p_1.

2. (ii)

   Find an equation, in scalar product form, for the plane p_2 that contains l_1 and the point B(0, -1, -2).

3. (iii)

   Find an equation for the line of intersection, l_2, between p_1 and p_2.

4. (iv)

   A plane, p_3, has an equation $\mathbf{r} \cdot \begin{pmatrix}1\\a\\b\end{pmatrix} = 3.$ What is the relationship between a and b such that p_3 is parallel to l_2?