Condition for a Line to Lie in a Plane

A straight line lies wholly in a plane precisely when two conditions both hold: every point of the line lies in the plane, AND the plane's normal is perpendicular to the line's direction (so the line can't "tilt out" of the plane).

1. Vector form. The line $\vec r=\vec a+t\vec b$ lies in the plane $\vec r\cdot\vec n=d$ iff

   $\vec a\cdot\vec n=d\qquad\text{and}\qquad \vec b\cdot\vec n=0.$

   The first condition is "the line's base point is on the plane"; the second is "the line's direction is perpendicular to the plane's normal" (equivalently, parallel to the plane).
2. Cartesian form. The line $\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$ lies in the plane $Ax+By+Cz+D=0$ iff $Ax_1+By_1+Cz_1+D=0\qquad\text{and}\qquad aA+bB+cC=0.$ …