Write parametric equations of lines

A line is represented by a point and a nonzero direction vector, $\mathbf r(t)=\mathbf r_0+t\mathbf v$ , or component-wise as $x=x_0+at,\ y=y_0+bt,\ z=z_0+ct$ , with $t$ ranging over the real numbers. The direction vector can be obtained from two points, and different nonzero scalar multiples of it describe the same line; this scope is limited to straight lines in two- and three-dimensional coordinate geometry, not parametric curves or higher-dimensional generalizations.

Detailed Explanation: Write parametric equations of lines

A line can be written using:

a point on the line, and
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.

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3D Space - Line Vector Form - Two Points to Vector Form

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