Vector Equation of Straight Line

Theorem
Let $\mathbf a$ and $\mathbf b$ denote the position vectors of two points in space

Let $L$ be a straight line in space passing through $\mathbf a$ which is parallel to $\mathbf b$.

Let $\mathbf r$ be the position vector of an arbitrary point on $L$.

Then:
 * $\mathbf r = \mathbf a + t \mathbf b$

for some real number $t$, which may be positive or negative, or even $0$ if $\mathbf r = \mathbf a$.

Proof

 * Vector-equation-of-straight-line.png

Let $a$ and $b$ be points as given, with their position vectors $\mathbf a$ and $\mathbf b$ respectively.

Let $P$ be an arbitrary point on the straight line $L$ passing through $\mathbf a$ which is parallel to $\mathbf b$.

By the parallel postulate, $L$ exists and is unique.

Let $\mathbf r$ be the position vector of $P$.

Let $\mathbf r = \mathbf a + \mathbf x$ for some $\mathbf x$.

Then we have:
 * $\mathbf x = \mathbf r - \mathbf a$

As $\mathbf a$ and $\mathbf r$ are both on $L$, it follows that $\mathbf x$ is parallel to $\mathbf b$.

That is:
 * $\mathbf x = t \mathbf b$

for some real number $t$.

Hence the result.