Two walls meeting make a corner line: Two non-parallel planes share a whole line — think of two walls of a room meeting in a vertical edge. To find it you need a point on it and a direction.
Building the line
Direction
The line lies in BOTH planes, so it's perpendicular to BOTH normals — that's exactly the cross product d = n₁ × n₂.
Point
Pick a convenient value for one coordinate (often z = 0), then solve the two plane equations as a simple 2×2 system for the other two.
IB-style question — find the line of intersection
Two planes have equations Π₁: x + y + z = 6 and Π₂: x − y + 2z = 5.
Find a vector equation of their line of intersection.
Step by step
- Read off the normals from the coefficients.
- Direction = n₁ × n₂. Compute each component (i, j, k).
- Simplify the three components.
- Find one point: set z = 0, giving x + y = 6 and x − y = 5.
- Add the two equations: 2x = 11 ⇒ x = 5.5, then y = 0.5.
- Write the line with that point and the direction.
Final answer
r = (11/2, 1/2, 0) + λ(3, −1, −2). (Any point on the line and any nonzero multiple of (3, −1, −2) is acceptable.)
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Watch what the angle is measured FROM: Two planes are measured through their normals. A line is measured to the plane's flat surface, so the formula changes from cosine to sine.
Angle between two planes
- = angle between their normals.
- cos θ = |n₁·n₂| / (|n₁||n₂|).
- We take the absolute value so we report the acute angle.
Angle between a line and a plane
- The angle to the flat surface, not to the normal.
- sin θ = |d·n| / (|d||n|).
- Use the line's direction d and the plane's normal n.
Why sine: The normal sticks up at 90° to the surface, so the line-to-surface angle and the line-to-normal angle add to 90° — that swaps cosine for sine.
IB-style question — angle between two planes
Find the acute angle between the planes Π₁: x + 2y + 2z = 1 and Π₂: 2x − 2y + z = 3.
Step by step
- Read off the normals.
- Dot product n₁·n₂.
- The dot product is 0, so the normals are perpendicular.
- cos θ = 0 means θ = 90°.
Final answer
The planes are perpendicular: the angle is 90°. (Here |n₁| = |n₂| = 3.)
IB-style question — angle between a line and a plane
Find the acute angle between the line r = (1, 0, 2) + λ(2, 1, 2) and the plane x + 2y + 2z = 7.
Step by step
- Direction of the line, and normal of the plane.
- Use SINE (line-to-surface). First d·n.
- Magnitudes: |d| = √(4+1+4) = 3, |n| = √(1+4+4) = 3.
- Apply the formula.
- Take the inverse sine.
Final answer
θ ≈ 62.7° (sin θ = 8/9). Using sine — not cosine — because the angle is to the surface.