Direction Numbers of a Line
Simple Explanation
A line in 3D can be described by any one point on it, plus a set of "direction numbers" (a,b,c) that indicate which way the line points — found from the coordinate differences between any two points on the line.
Why Do We Need It?
Direction numbers are the essential ingredient (along with one known point) for writing a line's equation, covered in the next concept.
Formula
Direction Numbers of a Line
a = x₂−x₁, b = y₂−y₁, c = z₂−z₁
Given two points on a line, the direction numbers (a,b,c) describe the line's direction — proportional to how far apart the two points are along each axis.
- (x₁,y₁,z₁), (x₂,y₂,z₂)
- — any two distinct points on the line
- a, b, c
- — the direction numbers of the line
When to use it: Whenever the direction of a line in 3D needs to be described, before writing its equation.
Worked Example
Find the direction numbers of a line
Find the direction numbers of the line through A(1,2,3) and B(4,0,7).
Why Does This Work?
Moving from A to B involves a specific change in each coordinate — these three changes ARE the direction the line travels; any point further along the same line is reached by adding a multiple of these same three changes, which is exactly what direction numbers capture.
Real-Life Example
Describing a flight path's heading
An air traffic controller knows a plane's position at two moments and needs to describe the straight-line direction it is heading.
The direction numbers between the two known positions give exactly this heading, in three-dimensional terms (accounting for altitude change too).
Practice
Find the direction number "b" for the line through (0,5,1) and (6,1,9).
MediumCommon mistake
Subtracting the coordinates in an inconsistent order across the three axes (e.g. B−A for x but A−B for y) — the SAME order must be used for all three coordinates.
Quick Review
- a = x₂−x₁, b = y₂−y₁, c = z₂−z₁.
- Direction numbers describe which way a line points, independent of which point is used.
- Use the same subtraction order for all three coordinates.