Definition:Inclination/Straight Line to Plane

Definition

Let $P$ be a plane.

Let $L$ be a straight line which intersects $P$ at the point $A$.

Let $Q$ be the perpendicular from a point $B$ on $L$.

Let $C$ be the point where $Q$ intersects $P$.

The inclination of $L$ to $P$ is defined as the angle $BAC$.

In the above diagram, this has been marked as $\theta$.

In the words of Euclid:

The inclination of a straight line to a plane is, assuming a perpendicular drawn from the extremity of the straight line which is elevated above the plane to the plane, and a straight line joined from the point thus arising to the extremity of the straight line which is in the plane, the angle contained by the straight line so drawn and the straight line standing up.

(The Elements: Book $\text{XI}$: Definition $5$)

Also known as

The inclination of a line to a plane can be variously described as:

the inclination of a line with a plane

the inclination of a plane with a line

the inclination of a plane to a line

the inclination between a plane and a line

and so on.

Some sources refer to an inclination as an angle. However, this lacks precision and can cause confusion.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): inclination: 2.
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): inclination