Distance between two points
In three-dimensional space:
d = √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²]
Example: A(1, 2, 3) and B(4, 6, 15): d = √(9 + 16 + 144) = √169 = 13
Distance from a point to a plane
The shortest distance is the perpendicular distance:
d = |Ax₀ + By₀ + Cz₀ + D| / √(A² + B² + C²)
Example: P(1, 2, 3) to the plane 2x − y + 2z − 5 = 0
- Numerator: |2 − 2 + 6 − 5| = 1
- Denominator: √(4 + 1 + 4) = 3
- d = 1/3
Distance between parallel planes
Pick any point on one plane and find its distance to the other.
Example: 2x + y − 2z = 4 and 2x + y − 2z = 10
- Point on the first: (2, 0, 0)
- Distance to the second: |4 − 10|/3 = 2
Distance from a point to a line (3D)
d = |PQ⃗ × m| / |m|, where Q is a point on the line and m is its direction.
Example: P(1, 1, 1); line through Q(0, 0, 0) with m = [1, 0, 0].
- PQ⃗ = Q − P = [−1, −1, −1]
- PQ⃗ × m = [0, −1, 1] → magnitude √2
- d = √2/1 ≈ 1.41
Angle between lines
Use their direction vectors:
cos θ = |m₁ · m₂| / (|m₁||m₂|)
(The absolute value gives the acute angle.)
Example: m₁ = [1, 2, 2], m₂ = [2, −1, 2]
- m₁ · m₂ = 2 − 2 + 4 = 4
- |m₁| = 3, |m₂| = 3
- cos θ = 4/9 → θ ≈ 63.6°
This is the angle between lines.
Angle between planes
The angle between planes equals the angle between their normal vectors.
Example: x + y = 3 and y + z = 1
- n₁ = [1, 1, 0], n₂ = [0, 1, 1]
- cos θ = 1/(√2 × √2) = 0.5 → θ = 60°
Angle between a line and a plane
If the line’s direction m makes angle φ with the plane’s normal n, the angle between the line and the plane is 90° − φ:
sin θ = |m · n|/(|m||n|)
Applications
- Clearance of a drone above a sloped roof (point-to-plane distance)
- Angles between roof panels or crystal faces
- GPS and 3D modelling
Worked examples
Example 1: Distance from (0, 0, 0) to x + 2y + 2z = 6? 6/3 = 2.
Example 2: Distance between (2, −1, 0) and (5, 3, 0)? 5.
Example 3: Are planes x + y + z = 1 and x − y = 4 perpendicular? n₁ · n₂ = 0 → yes (90°).
Common mistakes
- Forgetting the absolute value in the distance formula.
- Using the plane equation without moving everything to one side (Ax + By + Cz + D = 0).
- Giving an obtuse angle when the acute angle is wanted.
Try it yourself
- Distance between (1, −2, 4) and (3, 0, 5).
- Distance from (3, 1, −2) to 2x + 2y − z = 3.
- Angle between planes 2x − y + z = 0 and x + y + 2z = 5.
- Angle between lines with directions [1, 0, 0] and [1, 1, 0].
Answers: 1) 3 2) 7/3 3) 60° 4) 45°
Quick recap
- 3D distance extends Pythagoras to three coordinates.
- Perpendicular distance to a plane: |Ax₀ + By₀ + Cz₀ + D|/|n|.
- Angles between lines use direction vectors; angles between planes use normal vectors.