AHL 3.18 (HL)—Intersections and angles in 3D

3D intersections and angles come from shared equations.

An intersection is a point satisfying both objects; angles are then found from direction vectors or normals rather than from a misleading 2D sketch.

Example

Substitute a line into a plane equation to solve λ, then use the resulting point to verify the intersection and compute any requested angle.

Solve the shared condition first, check it in the original equations, and use the relevant dot-product angle formula only afterwards.

A line parallel to a plane has no intersection unless it lies in the plane; supplementary angle conventions must be stated.

Intersection workflow: substitute a line into a plane, or solve the simultaneous Cartesian equations for two or three planes, then interpret no solution, one solution or a family geometrically. If $\alpha$ is the acute angle between line direction $\mathbf d$ and plane normal $\mathbf n$, the line-plane angle is $90^\circ-\alpha$, equivalently $\sin\phi=|\mathbf d\cdot\mathbf n|/(|\mathbf d||\mathbf n|)$. For planes, use their normals in $\cos\theta=|\mathbf n_1\cdot\mathbf n_2|/(|\mathbf n_1||\mathbf n_2|)$.