3D Lines and Planes Calculator
Lines and planes in three dimensions: the line through two points, the angle and shortest distance between two lines, the plane through three points, the angle and line of intersection of two planes, and where a line meets a plane, with exact answers.
About the 3D Lines and Planes Calculator
A line is r = a + λb: a point a and a direction b. Two lines are parallel when their directions are multiples, intersecting when they meet, and skew when they're neither; the shortest distance between skew lines is |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|.
A plane is r · n = d: every point whose position vector has the same dot product with the normal n. The calculator also finds where a line meets a plane, the angle between them, and the line where two planes cross.
The formula
d = |(a₂ − a₁) · (b₁ × b₂)| ÷ |b₁ × b₂|
- cos θ
- |b₁ · b₂| ÷ (|b₁||b₂|)
- Plane through A, B, C
- n = AB × AC, n · r = n · A
- Two planes
- line of intersection along n₁ × n₂; cos θ = |n₁ · n₂| ÷ (|n₁||n₂|)
- Line and plane
- sin φ = |n · b| ÷ (|n||b|)
Worked example
r = (1, 2, 3) + λ(2, 3, 4) and r = (2, 4, 5) + μ(3, 4, 5)
- b₁ × b₂ = (−1, 2, −1), (a₂ − a₁) = (1, 2, 2)
- (1, 2, 2) · (−1, 2, −1) = 1
- d = 1/√6 = √6/6
Questions people ask
How do I know the lines are skew?
If b₁ × b₂ isn't 0 (not parallel) and the shortest distance isn't 0 (they don't meet), they're skew.
What are direction cosines?
The cosines of the angles a line makes with the x, y and z axes: the direction ratios divided by the line's length.