Plane equation through point and parallel to 2 lines

Question

We have the point $A(1,2,1)$ and the lines:

d1: $$x+2y-z+1 = 0 , x-y+z-1=0$$ d2: $$2x-y+z=0, x-y+z=0$$

Write the equation of the plane that passes through $A$ and is parallel to the two lines.

I'm really lost on this one. Got no clue whatsoever on what I have to do here.

Answer 1

Here is an outline. I am renaming your known point to $P$ so I can use the standard notation in my step 4.

1. Find parametric equations for each line. One way to do that is to take $z=t$ then treat $t$ as a constant. That gives you two equations in two unknowns $x$ and $y$ which you can solve by the usual techniques. You will get linear expressions for $x$ and $y$ in terms of $t$, which are your parametric equations.
2. From those parametric equations, get a vector parallel to each line. Just use the coefficients in front of $t$ in the expressions for $x$, $y$, and $z$.
3. Find the cross-product of those two vectors. That vector, perpendicular to the two lines, will also be perpendicular to your desired plane, so it is a normal vector for the plane.
4. The coordinates of that cross-product are the coefficients of the general linear equation $Ax+By+Cz=D$ for your desired plane. Substitute the coordinates of your known point $P$ to find the constant $D$. Now you are done.

Let me know if you have trouble in any of those steps.