Find the general form equation of a plane that contains the point $(2, -2, 1)$ and passes through a line $(x, y, z) = (1, 2, -3) + t(2, -3, 2)$.
I managed to use the fact that it contains a given point to write the equation of the plane as $A(x-2) + B(y+2) + C(z-1) = 0$ but got stuck there.
Best Answer
Actually you want to use the point $(2, -2, 1)$ to set up the equation, so you can use the line to solve it: $$A(x-2) + B(y+2) + C(z-1) = 0$$ and then solve $$A((1+2t)-2) + B((2-3t)+2) + C((-3+2t)-1) = 0$$ by plugging in some useful values of $t$. So $$A(2t-1) + B(-3t+4) + C(2t-4) = 0$$ and try out $t=\frac{1}{2}$, then $t=\frac{4}{3}$, then $t=2$ and solve for $A,B,C$.