Skill: Write parametric equations of lines

A line can be written using:

1. a point on the line, and
2. a nonzero direction vector showing how the line moves.

The vector form is

$$\mathbf r(t)=\mathbf r_0+t\mathbf v,\qquad t\in\mathbb R,$$

where $\mathbf r_0$ is a point on the line and $\mathbf v$ is its direction vector.

Example

Find parametric equations for the line through

$$A(1,2,-1)\quad\text{and}\quad B(4,0,5).$$

Step 1: Find a direction vector.

Subtract the coordinates of $A$ from those of $B$:

$$\overrightarrow{AB}=B-A =(4-1,\ 0-2,\ 5-(-1)) =(3,-2,6).$$

So a direction vector is

$$\mathbf v=(3,-2,6).$$

Step 2: Choose a point on the line.

Use $A(1,2,-1)$ as the starting point:

$$\mathbf r_0=(1,2,-1).$$

Step 3: Substitute into the line formula.

$$\mathbf r(t)=(1,2,-1)+t(3,-2,6), \qquad t\in\mathbb R.$$

Therefore, the parametric equations are

$$\boxed{x=1+3t,\qquad y=2-2t,\qquad z=-1+6t,\qquad t\in\mathbb R.}$$

When $t=0$, the equations give the point $A(1,2,-1)$. When $t=1$, they give

$$(1+3,\ 2-2,\ -1+6)=(4,0,5),$$

which is point $B$. Thus the equations describe the entire line through $A$ and $B$.

Any nonzero scalar multiple of the direction vector, such as $(6,-4,12)$, would describe the same line.