Get the general equation of the plane $\pi$ in each case

Question

a) $\pi$ contains $P=(1,0,-1)$ and $r_{.}(x-2)/2=y/3=2-z$

b) $\pi$ contains $P=(1,-1,1)$ and $r_{.}X=(0,2,2)+ \lambda(1,1,-1)$

I'm trying to remember vectors in the plane, but I don't remember any

Answer 1

A plane containing a normal vector $\mathbf{N}=\langle a,b,c\rangle$ and containing a point $P_0=(x_0,y_0,z_0)$ which is not on the line has an equation

$$ a(x-x_0)+b(y-y_0)+c(z-z_0)=0 $$

In each of the exercises, you are given such a point $P_0$. So you must, in each case, find a normal vector $\mathbf{N}$.

In each case, instead of being given a normal vector directly, you are given information from which you can find a normal vector; you are given the equation of a line that lies in the plane.

For each of the two exercises, find coordinates of two points $A$ and $B$ which lie on the line. Then a normal vector for the plane will be the cross-product of the two vectors $\vec{P_0A}$ and $\vec{P_0B}$

$$ \mathbf{N}=\vec{P_0A}\times\vec{P_0B} $$

You should be able to find points $A$ and $B$ yourself to complete the exercises.