Plane containing a line and a point

Find the plane containing line r (through A(8, 2, 6), direction (6, 6, 3)) and point P(19, 11, 7).

Step-by-step solution

Join A with P: that line and r determine the plane.
The normal is the cross product of the directions of the two lines.
If P lay on r, both directions would coincide and there would be infinitely many planes: it would not be determined.
Check that A and P satisfy the plane equation.

Result: α: −21x + 27y − 12z + 186 = 0.

A line and an outside point determine a plane; if the point lies on the line, they do not.