How do I find equation of a parallel plane given with 2 points.

Question

Find the equation of plane $W: (Ax+By+Cz+D=0)$ that passes through point $M(-3,-1,0)$ and $N(0,1,2)$ and its normal to the plane $W_1: 2x-y+2z-1=0$.
I know that if two planes are normal then their vector are normal. So we can say that $AA_1+BB_1+CC_1=0$.
And I know how to find vector from $MN(x_2-x_1,y_2-y_1,z_2-z_1)$. I have no idea how to continue. Anyone can help please...

Answer 1

HINT

• from normal condition we obtain $2A-B+2C=0$

• then use the also 2 condition of passage through points M and N to obtain other 2 equations

Note that we obtain a system of three equations in 4 unknowns but one among A,B,C,D can arbitrarily fixed.

Answer 2

Using the conditions of your problem we get the equations: $$-3A-B+D=0$$ $$B+2C+D=0$$ $$2A-B+2C=0$$