How to find vector and parametric equations of a line segment

If you know the two endpoints of a line segment, you can describe every point in between with one parameter. We’ll write the vector equation first, then read the three parametric equations directly from its coordinates.

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Use the endpoints to write a vector equation

Let ${\mathbf{r}}_0$ and ${\mathbf{r}}_1$ be the position vectors of the first and second endpoints. To reach a point on the segment, start at ${\mathbf{r}}_0$ and move a fraction $t$ of the way toward ${\mathbf{r}}_1$.

$\begin{array}{l}\mathbf{r}\left(t\right)={\mathbf{r}}_0+t({\mathbf{r}}_1-{\mathbf{r}}_0)\\ \mathbf{r}\left(t\right)=(1-t){\mathbf{r}}_0+t{\mathbf{r}}_1\\ 0\le t\le 1\end{array}$

At $t=0$ we get the first endpoint, and at $t=1$ we get the second. Values between them trace the segment, including its midpoint at $t=1/2$.

Substitute the two points and combine components

Let’s use the endpoints $P(1,2,-1)$ and $Q(1,0,3)$. Their coordinates give us the two position vectors.

$\begin{array}{l}{\mathbf{r}}_0=⟨1,2,-1⟩\\ {\mathbf{r}}_1=⟨1,0,3⟩\end{array}$

Substitute the vectors into the formula. Multiply each vector by its scalar, then add the matching components.

$\begin{array}{l}\mathbf{r}\left(t\right)=(1-t)⟨1,2,-1⟩\\ +t⟨1,0,3⟩\\ \mathbf{r}\left(t\right)=⟨1,2-2t,-1+4t⟩\end{array}$

In unit-vector notation, the three components are the coefficients of $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$.

$\begin{array}{l}\mathbf{r}\left(t\right)=\mathbf{i}+(2-2t)\mathbf{j}\\ +(-1+4t)\mathbf{k}\end{array}$

Read off the parametric equations and check the endpoints

The first component is $x$, the second is $y$, and the third is $z$. Each coordinate is a function of the same parameter $t$.

$\begin{array}{l}x=1\\ y=2-2t\\ z=-1+4t\\ 0\le t\le 1\end{array}$

Plugging in $t=0$ gives $(1,2,-1)$, and plugging in $t=1$ gives $(1,0,3)$. The $x$-coordinate stays at $1$ throughout, so the whole segment lies in the plane $x=1$.

As t runs from 0 to 1, the point goes from P through the midpoint to Q.

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