Describing a line with vectors
A line is fixed by one point on it and one direction.
Vector equation of a line:
r = r₀ + t m
- r₀ = position vector of a known point
- m = direction vector (parallel to the line)
- t = parameter (any real number)
Each value of t gives a point on the line.
Example in 2D
Line through A(2, 1) with direction m = [3, −2]:
[x, y] = [2, 1] + t[3, −2]
| t | Point |
|---|---|
| 0 | (2, 1) |
| 1 | (5, −1) |
| −1 | (−1, 3) |
Parametric equations
Split the vector equation into parametric equations:
- x = 2 + 3t
- y = 1 − 2t
Line through two points
Use one point and the vector between the points as m.
Example: Through P(1, 4, −2) and Q(3, 0, 5)
- m = PQ⃗ = [2, −4, 7]
- [x, y, z] = [1, 4, −2] + t[2, −4, 7]
Parametric form:
- x = 1 + 2t
- y = 4 − 4t
- z = −2 + 7t
Symmetric equations (3D)
Solve each parametric equation for t and set them equal:
(x − 1)/2 = (y − 4)/(−4) = (z + 2)/7
These are the symmetric equations (only possible when no component of m is 0).
Scalar equation (2D only)
In 2D, lines can also be written as Ax + By + C = 0, the scalar equation. The vector [A, B] is perpendicular (normal) to the line.
From x = 2 + 3t, y = 1 − 2t: eliminate t → 2x + 3y − 7 = 0.
Is a point on the line?
Substitute and check whether one value of t works for every coordinate.
Example: Is (7, −8, 19) on the 3D line above?
- x: 1 + 2t = 7 → t = 3
- y: 4 − 4(3) = −8 ✓
- z: −2 + 7(3) = 19 ✓ → yes
Parallel and identical lines
- Parallel if their direction vectors are scalar multiples.
- The same line if they’re parallel and share a point.
Worked examples
Example 1: Vector equation through (0, 3) with direction [1, 2]: [x, y] = [0, 3] + t[1, 2].
Example 2: Direction vector of x = 5 − t, y = 2 + 4t? [−1, 4].
Example 3: Are [x, y] = [1, 1] + t[2, 4] and [x, y] = [0, 5] + s[1, 2] parallel? [2, 4] = 2[1, 2] → yes.
Common mistakes
- Using a point as the direction vector.
- Using different t-values for different coordinates when testing a point.
- Dividing by zero in symmetric form when a component is 0.
Try it yourself
- Vector equation through A(4, −1) and B(6, 3).
- Parametric equations for [x, y, z] = [2, 0, 1] + t[1, −3, 2].
- Is (3, −3, 3) on the line in question 2?
- Write the line in question 1 as a scalar equation.
Answers: 1) [x, y] = [4, −1] + t[2, 4] 2) x = 2 + t, y = −3t, z = 1 + 2t 3) yes (t = 1) 4) 2x − y − 9 = 0
Quick recap
- Vector equation of a line: r = r₀ + t m with a direction vector m.
- Convert to parametric equations, symmetric equations (3D) or a scalar equation (2D).
- Test points by finding a single value of the parameter.