Relative positions of line and plane and angles

In brief — Line and plane

Given the line $r$ with direction $\vec v$ and the plane $\pi$ with normal $\vec n$:

• Coincident: the line lies in the plane (identity in the system).
• Incident: a single point of intersection; the line–plane angle satisfies $\sin\alpha = \dfrac{|\vec v\cdot\vec n|}{|\vec v|\,|\vec n|}$.
• Parallel: no common point ($\vec v\cdot\vec n=0$ but $P_0\notin\pi$).

The dihedral angle between two planes is the angle between their normal vectors: $\cos\theta = \dfrac{|\vec n_1\cdot\vec n_2|}{|\vec n_1|\,|\vec n_2|}$.

Mind the difference: for the line–plane angle the sine appears (the line forms with the plane the angle complementary to the one with the normal), whereas for the angle between two planes the cosine of the normal vectors appears.

The line $r$ is incident on the plane $\pi$; $\alpha$ is the line–plane angle, $\vec n$ the normal.

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Topics: Analytic geometry in space
Concepts: Dihedral angle · Plane · Dot product · Line · Line in space
Skills: Analytic geometry · Reasoning by cases · Using formulae