Intersection of a line and plane, angle between line and plane, intersection and angle between two planes, intersection of 3 planes
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Practice exam-style angles and intersections with planes problems
To find the intersection of a line and a plane, substitute the vector line equation into the plane equation and solve for the parameter λ.
The angle ϕ between a line and a plane is measured as the complement of the angle between the line’s direction vector d and the plane’s normal vector n.
Since the normal forms an angle of 90° with the plane, we can construct a right angled triangle using the intersection of the normal, the line, and the plane:
If θ is the angle between d and n, then ϕ=180°−90∘−θ=90°−θ
In practice, you can compute θ using cosθ=∣d∣∣n∣d⋅n.
The intersection of 2 planes - if they are not parallel - is a line. We can find the line intersection by solving a system of equations.
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The angle θ between two planes can be found by computing the angle between the two plane normals,
When three planes are considered together, their intersection in 3D space can take several forms:
A plane if all three planes coincide.
A single point if the three planes intersect uniquely, meaning their normal vectors are not all parallel or in some degenerate arrangement, and the system of equations has exactly one solution.
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A line if two planes intersect in a line and the third plane also contains that line (or if each pair of planes meets along the same line).
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No intersection if the system of equations is inconsistent (e.g., the planes are arranged in parallel or partially parallel ways that do not share a common point).
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