Find the equations of the following plane in both cartesian and parametric form:
The plane through the point $(1,4,5)$ and perpendicular to the vector $(7,1,4)$.
So far, I have obtained the cartesian form, which is: $$7x + 4y + z =31.$$
How do I change this to the parametric form: $r = r_0 + s\vec u + t\vec v$ where $s$ and $t$ are real numbers?
you need to find two more points which lie on the plane, for example $B(4,0,3)$ and $C(0,7,3)$ then your parametric equation will be $$\underline{r}=\left(\begin{matrix}1\\4\\5\end{matrix}\right)+\lambda\left[\left(\begin{matrix}1\\4\\5\end{matrix}\right)-\left(\begin{matrix}4\\0\\3\end{matrix}\right)\right]+\mu\left[\left(\begin{matrix}1\\4\\5\end{matrix}\right)-\left(\begin{matrix}0\\7\\3\end{matrix}\right)\right]$$ $$\Rightarrow \underline{r}=\left(\begin{matrix}1\\4\\5\end{matrix}\right)+\lambda\left(\begin{matrix}-3\\4\\2\end{matrix}\right)+\mu\left(\begin{matrix}1\\-3\\2\end{matrix}\right)$$
