Since 2 lines define a plane, the linear span is a plane defined by the lines and . To make the plane pass through a point P, one needs only to add the vector P to every vector in .
A set M of points is called a plane if there exists a point P and 2 L.I. vectors A and B such that
More specifically, M is called a plane through P spanned by A and B.
Two planes and through the same point P are equal iff .
But the definition of a plane, it is obvious that Þ .
Conversely, if , then for any given s and t, there exist s' and t' such that
i.e., . Q.E.D.
Two planes and spanned by the same vectors A and B are equal iff Q is on M.
Two planes and are said to be parallel if . Also, a vector X is parallel to the plane M if .
Given a point Q not on plane M, there is one and only one plane M' that contains Q and parallel to M.
Q, R be 3 points not on the same line.
There is one and only one plane M
containing these 3 points.
3 vectors A, B, C are L.I. iff they lie on the same plane through O.