Line perpendicular to a plane

In space, perpendicularity no longer concerns only two lines, but also a line and a plane. The idea is that the line be “straight” with respect to the plane in every direction.

Definition — Line perpendicular to a plane

A line $r$ is perpendicular to a plane $\pi$ if it intersects $\pi$ at a point $P$ and is perpendicular to every line of the plane passing through $P$.

Checking perpendicularity with respect to all the lines of the plane would be impossible in practice. Fortunately two are enough.

Theorem — Line-plane perpendicularity criterion

A line $r$ is $\perp$ to a plane $\pi$ at $P$ if and only if it is $\perp$ to two lines of $\pi$ passing through $P$ and not parallel to each other. There is no need to check all the lines of the plane: two are enough.

Links

Topics: Synthetic geometry in space
Concepts: Line-plane perpendicularity
Skills: Synthetic geometry