Skill: Write vector equations of lines

A vector equation for a line has the form

$$\mathbf r=\mathbf a+t\mathbf d,$$

where:

• $\mathbf r$ is the position vector of any point on the line,
• $\mathbf a$ is the position vector of a known point on the line,
• $\mathbf d$ is a nonzero direction vector,
• $t\in\mathbb R$.

When two points are given, subtract their coordinates to find a direction vector.

Example

Find a vector equation for the line through

$$A(2,-1,3) \quad\text{and}\quad B(5,1,-3).$$

First, use one of the points as the known point. Take $A$:

$$\mathbf a= \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix}.$$

Next, find a direction vector by subtracting the coordinates:

$$\overrightarrow{AB}=B-A = \begin{pmatrix} 5\\ 1\\ -3 \end{pmatrix} - \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix} = \begin{pmatrix} 3\\ 2\\ -6 \end{pmatrix}.$$

Therefore, the vector equation of the line is

$$\boxed{ \mathbf r= \begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix} +t \begin{pmatrix} 3\\ 2\\ -6 \end{pmatrix}, \qquad t\in\mathbb R }$$

The value $t=0$ gives point $A$. The value $t=1$ gives

$$\begin{pmatrix} 2\\ -1\\ 3 \end{pmatrix} + \begin{pmatrix} 3\\ 2\\ -6 \end{pmatrix} = \begin{pmatrix} 5\\ 1\\ -3 \end{pmatrix},$$

which is point $B$. Since $t$ can be any real number, the equation describes the entire line.