# Definition:Inclination/Straight Line to Plane

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## 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): Entry:**inclination**:**2.** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**inclination**