Recipe

Write parametric and symmetric equations of a line in space

Use when: you're given a point and a direction vector, or two points, and need the line through them as parametric or symmetric equations

Goal: parametric equations for x, y, z, and, when no direction component is zero, the matching symmetric equations

Taught in Lines in space

Quick recipe

Get a point and a direction vector
Write the parametric equations
Convert to symmetric equations, when every direction component is nonzero
Check the equations against a known point

1
A line's equations always start from one point and one direction, but a problem can hand those over in two different shapes.

Example

Given $P_1=(1,0,-2)$ and $P_2=(3,5,-1)$ : $\mathbf v=\langle3-1,\,5-0,\,-1-(-2)\rangle=\langle2,5,1\rangle$ . Take $P_0=P_1=(1,0,-2)$ .
- a point $P_0$ and a direction vector $\mathbf v$ are given directly — use them as they are.
- two points $P_1,P_2$ are given instead — compute the direction as $\mathbf v=\overrightarrow{P_1P_2}$ , subtracting matching coordinates, and use either point as $P_0$ .
Why: every later step needs exactly one point and one direction vector, whichever form the problem originally handed over.
2
Read the point's coordinates and the direction's components straight into $x=x_0+at,\ y=y_0+bt,\ z=z_0+ct$ .

Example

$x=1+2t,\quad y=5t,\quad z=-2+t$ .

Why: these three equations are exactly the vector equation $\mathbf r(t)=\mathbf r_0+t\mathbf v$ , read off one component at a time.
3
Look back at the direction vector's three components from Step 1 before dividing by any of them.

Example

With $\mathbf v=\langle2,5,1\rangle$ , none of the components are zero, so $\dfrac{x-1}{2}=\dfrac{y}{5}=\dfrac{z+2}{1}$ .
- $a,b,c$ are all nonzero — solve each parametric equation for $t$ and set the results equal, $\dfrac{x-x_0}{a}=\dfrac{y-y_0}{b}=\dfrac{z-z_0}{c}$ .
- one of $a,b,c$ is zero — the symmetric form doesn't exist; report the parametric equations only, and give that one coordinate's fixed value directly alongside the remaining ratio.
Why: dividing by a zero direction component is undefined, so symmetric form only exists for a line that isn't parallel to any coordinate plane.
4
Substitute a specific value of $t$ and confirm the resulting point matches a point the line was supposed to pass through.

Example

At $t=1$ : $x=1+2=3,\ y=5,\ z=-2+1=-1$ , matching $P_2=(3,5,-1)$ exactly.

Why: this catches an arithmetic slip in the direction vector or the point before the equations get used anywhere downstream.

Common mistakes

Subtracting the two points in the wrong order, which still describes the same line but can look wrong against an answer key.
Writing symmetric equations when a direction component is zero, dividing by zero without noticing.
Using a point that isn't actually on the line as $P_0$ .
Confusing the direction vector's own components with a position vector, when the direction vector's tail location never matters.