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