Vector Equation of Straight Line in Space
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
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.
$\blacksquare$
Sources
- 1927: C.E. Weatherburn: Differential Geometry of Three Dimensions: Volume $\text { I }$ ... (previous) ... (next): Introduction: Vector Notation and Formulae
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $3$. Addition and Subtraction of Vectors
- 1957: D.E. Rutherford: Vector Methods (9th ed.) ... (previous) ... (next): Chapter $\text I$: Vector Algebra: $\S 9$: $(17)$