Problem

The planes $\Pi_1$ and $\Pi_2$ do not intersect and are both perpendicular to $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$. The line $l$ intersects $\Pi_1$ at the point $(1,6,0)$ and intersects $\Pi_2$ at the point $(3,-6,0)$.

(a) Find Cartesian equations of $\Pi_1$ and $\Pi_2$.

(b) Express the vector equation of $l$ in the form

$$\begin{pmatrix}x\\y\\z\end{pmatrix}=\mathbf{a}+\lambda\mathbf{b},$$

where $\mathbf{a}$ and $\mathbf{b}$ are vectors to be determined, and hence show that for points on $l$,

$$\frac{1}{2}x+\frac{1}{12}y=1\qquad\text{and}\qquad z=0.$$