How to find equation of a plane given that it must contain a particular line and be perpendicular to another plane

Question

Suppose that I need to find the equation of the plane $P$ which contains a line $l(t)$, given parametrically as

$$ l(t) = \bigg(x(t), y(t), z(t) \bigg) = (v_1t + x_0, v_2t + y_0, v_3t + z_0). $$

The plane $P$ must also be perpendicular to another plane, whose equation is $Ax + By + Cz + D = 0$. Assuming that for given values of $v_1, v_2, v_3, x_0, y_0, z_0, A,B,C$ and $D$ this problem has a solution (a plane $P$ satisfying all of these required conditions), how would one go about starting to solve this problem?

Answer 1

The normal $N$ of your desired plane must be perpendicular to two vectors

1. The direction vector of the line, which is $(v_1, v_2,v_3)$
2. The normal vector of the given plane, which is parallel to $(A,B,C)$

Since it's perpendicular to these two vectors, $N$ must be parallel to their cross product.

Is that enough to get you started?