determine where a vector will intersect a plane

An outline of one method to find the point of intersection:

First find the equations of the line and the plane

A parameterization of the line is $$\tag{1} (x,y,z)= (o_1+d_1 t\, , o_2+d_2 t\,, o_3+d_3 t ),\quad -\infty<t<\infty. $$

To find an equation of the plane, take the cross product of the vectors $A-B$ and $B-C$. This will give you a normal vector to the plane: $(N_1, N_2, N_3)$. The equation of the plane is then, using $A$ as a point on the plane: $$\tag{2} N_1(x-a_1)+N_2(y-a_2)+N_3(z-a_3)=0. $$

Now, to find the point of intersection, substitute the information from $(1)$ $$ x=o_1+d_1 t , \quad y= o_2+d_2 t\quad z= o_3+d_3 t $$ into $(2)$ and solve for $t$. Then substitute this value of $t$ into $(1)$ to find the coordinates of the point.

I'm assuming there is a point of intersection. There may not be, or there may be infinitely many...