Derive the condition for the line…

Derive the condition for the line $\tfrac{x-x_1}{l}=\tfrac{y-y_1}{m}=\tfrac{z-z_1}{n}$ to lie in the plane $Ax+By+Cz+D=0$.

Step 1: Derivation

Substitute the general point $(x_1+lt,\ y_1+mt,\ z_1+nt)$: $(Ax_1+By_1+Cz_1+D)+t\,(Al+Bm+Cn)=0$.

This holds for every $t$ if and only if both the constant term and the coefficient of $t$ vanish: $Ax_1+By_1+Cz_1+D=0\qquad\text{and}\qquad Al+Bm+Cn=0$.

The first says $M_1$ lies in the plane. The second says $\vec d\perp\vec n$, i.e. the line is parallel to the plane. In vector form: $\vec n\cdot\vec r_1+D=0$ and $\vec n\cdot\vec d=0$.