The Vector Equation of a Line in Space
Simple Explanation
A line through a known point (with position vector r₀) in the direction of vector v can be written compactly as r = r₀ + t·v, where r is the position vector of any point on the line and t is a parameter.
Why Do We Need It?
This is the vector-notation counterpart of the parametric equations from the previous chapter — often a cleaner, more compact way to work with lines, especially in vector-based calculations.
Formula
The Vector Equation of a Line
r = r₀ + t·v
A line through the point with position vector r₀, in the direction of vector v, is described by this equation — every value of t gives a different point on the line.
- r
- — the position vector of a general point on the line
- r₀
- — the position vector of a known point on the line
- v
- — the direction vector of the line
- t
- — the parameter
When to use it: Whenever a line in 3D needs to be described using vector notation.
Worked Example
Write a line's vector equation
Write the vector equation of the line through the point with position vector r₀=(1,2,3), in the direction of v=(2,−1,4).
Why Does This Work?
Starting at the known point r₀ and adding t·v for different values of t traces out every point on the line — positive t moves in the direction of v, negative t moves the opposite way, and t=0 gives back the starting point exactly.
Real-Life Example
Modeling a laser beam's straight-line path
A laser starts at a known position and travels in a fixed direction through space.
The vector equation of a line gives the laser's exact position at any "time" t along its path.
Practice
For the line r=(1+2t, 2−t, 3+4t), find the y-coordinate at t=5.
MediumCommon mistake
Confusing r₀ (a fixed point the line passes through) with v (the direction the line travels) — they play very different roles in the equation.
Quick Review
- r = r₀ + t·v.
- r₀ is a known point on the line; v is the direction vector.
- Each value of t gives one specific point on the line.