Two vectors A and B are parallel iff for some .
Two lines and are equal iff A and B are parallel.
Let . A point X on the line can therefore be written as
Hence, where . That is, A and B are parallel.
Let A and B be parallel so that for some .
If Q is on , then for some . Thus,
Hence, Q is also on . Since this is true for arbitrary Q, we have . Reversing the role of A and B gives . Hence, .
Two lines and are parallel if A and B are parallel.
Given a line L and a point Q not on L, there is one and only one line L' through Q and parallel to L.
Let A be the direction vector of L. We can write . According to theorem 2.1, L' is unique. Q.E.D.
If , there is 1 and only 1 line containing both P and Q.
The line can be written as .
The line goes through both P and Q since they corresponds to and , respectively.
Let L' be another line that goes through both P and Q. Since L' goes through P, we have for some nonzero A. Now, L' also goes through Q so that for some α. Hence, so that A and Q-P are parallel. By theorem 2.2, . Q.E.D.
From theoem 2.4, we see that Q is on iff A and Q-P are parallel, i.e., .
Consider the 3-space case with and .
To test if is on , we examine and find that it is not a multiple of A. Hence, Q is not on the line.
For , we have so that Q is on L.
Two vectors are linearly dependent iff they lie on the same line through the origin.
If either vector is zero, the theorem holds trivially.
If both are nonzero, they are dependent iff for some .
However, iff B is on the line . Q.E.D.