- Prove that are all on the same line if and only if
- For two vectors with , prove that the line through and is the set of all vectors such that .

*Proof.*Assume all lie on a line , and let . ThenSo, we have

Thus, is the vector

Conversely, assume . From Theorem 13.12(g) we know this is the case if and only if the vectors and are linearly dependent. Hence, there are nonzero such that

Since is nonzero we may divide by , and we obtain

But this means is on the line passing through and

*Proof.*If , then we know there is a unique line passing through and , sayThus, any point is given by

Therefore, we have