Problem

The plane $\Pi_1$ has vector equation

$$\mathbf r= \begin{pmatrix}5\\3\\0\end{pmatrix} +s\begin{pmatrix}3\\0\\1\end{pmatrix} +t\begin{pmatrix}1\\-2\\2\end{pmatrix}$$

where $s$ and $t$ are scalar parameters.

(a) Determine a Cartesian equation for $\Pi_1$

The plane $\Pi_2$ has vector equation

$$\mathbf r\cdot\begin{pmatrix}5\\-2\\3\end{pmatrix}=1$$

(b) Determine a vector equation for the line of intersection of $\Pi_1$ and $\Pi_2$

Give your answer in the form $\mathbf r=\mathbf a+\lambda\mathbf b$, where $\mathbf a$ and $\mathbf b$ are constant vectors and $\lambda$ is a scalar parameter.

The plane $\Pi_3$ has Cartesian equation $4x-3y-z=0$

(c) Use the answer to part (b) to determine the coordinates of the point of intersection of $\Pi_1$, $\Pi_2$ and $\Pi_3$