Prove that the intersection of a line and a plane such that the line is not parallel to the plane contains one and only one point.
Proof. Denote the line by and the plane by . Let be the set of points
Since is not parallel to w know that its direction vector is not in the span of and . Further, by definition of a plane, we know the vectors and are linearly independent. Hence, are linearly independent. Then, any point in the intersection must be a solution to the system of equations
By the linear independence of we know this system has exactly one solution . Hence, contains exactly one point