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# Perpendicular Planes|Definition & Meaning

## Definition

**Planes** that are **perpendicular** to one **another** meet at a **90Â°** angle. Two **planes** are **perpendicular** to one other if and only if they **contain** a line that is **perpendicular** to each other. A line **perpendicular** to a plane **allows** for the **creation** of multiple other **planes** that are also **perpendicular** to the **original** plane. It follows that if a line is **perpendicular** to a plane, then any **lines** in the plane that cross that line are also **perpendicular** to the plane.

In **geometry,** a plane is a flat, **two-dimensional surface** that extends **indefinitely** in all **directions.** A plane is defined by three points that are not on the **same** line. The **points** can be **connected** to form a triangle, and the **plane** is the surface that contains the **triangle.Â **

A **plane** can be perpendicular to **another** plane, **meaning** that the two planes **intersect** at a right angle. When this **happens,** the planes are said to be **orthogonal** to each **other.**

To **visualize** this, imagine two **planes** that **intersect** at a right angle. If you were to draw a line on one plane and **extend** it until it **intersects** the other plane, the line would be **perpendicular** to the **second plane.**

**Perpendicular** planes are often used in **geometry** to describe the **orientation** of objects in **space.** For example, if a plane is perpendicular to the ground, it is said to be **“upright.” Similarly,** if a plane is **perpendicular** to a wall, it is said to be **“vertical.”**

## Understanding Perpendicular Planes Visually

If **line** l is also **perpendicular** to every one of the lines in plane A in which it intersects, then it can be said to be **perpendicular** to plane A.

**Consider** the image of a stick placed vertically on a plain and even surface. The **stick** is positioned in such a way that it is **perpendicular** to those lines that have been drawn on the **table** and that go through the **location** at which the stick is currently **standing.**

If plane B **includes** a line that also runs **perpendicular** to **plane** A, then **plane** B can be considered to be **perpendicular** to plane A.

The **following** figure shows the **illustration** of **perpendicular** planes.

**Planes** are considered **perpendicular** to one another if their **intersection** is at a **90-degree** angle. In geometry, one can encounter various distinct kinds of **perpendicular** planes, **including** the **following**

## Planes That Cross One Another

**Planes** that **cross** are not parallel to one **another,** and their points of **intersection** are always aligned along a line. It is **impossible** for two planes to **meet** in much more than a line at any given point.

The **diagram** that follows **illustrates** how two planes, P and Q, **meet** at a single point on plane XY. As a **result,** line PQ is the line that connects to the Q **plane** through the XY plane. **Intersecting** planes can be seen in action in the **form** of the two walls that **connect** the rooms.

## Orthogonal Lines or Planes

2 lines or indeed planes are said to be orthogonal if they intersect each other at right angles (90 **degrees).**

The lines AB and **PQ** in the illustration that follows are considered to be orthogonal because they **intersect** each other at a straight angle. In the context of geometry, the phrase “at right **angles”** is what the word “orthogonal” refers to.

**Additionally,** there are occasions when we refer to them as being “normal” to one another. To be very clear, the lines are not required to connect with one another at any point physically.

**Even** if they do not **intersect,** it is possible for two line segments to be considered orthogonal to one **another.** All that is required is that they are perpendicular to one another.

## Vertical Planes and Their Uses

A **plane** that is vertical is a **plane** that really is **perpendicular** to the surface of the **ground.** To put it another way, this is a plane that cuts across the **horizontal** plane at right angles.

It is possible to **define** a **vertical** plane in a **three-dimensional** coordinate system as a plane that is parallel to an x-y plane but perpendicular to the **z-axis.** This type of plane is known as a vertical plane.

For **instance,** the plane defined by the **equation** z = 0 is considered a vertical plane since it lies in a direction that is perpendicular to the x-y **plane,** which stands for the surface of the earth.

When **creating** a 3D model or **drawing,** vertical **planes** are **frequently** employed as the **representation** of walls, floors, as well as other vertical **surfaces.**

For the purpose of **illustrating** the wall of a **building, vertical** planes are frequently utilized in the field of **architecture.** In the field of engineering, the side of a bridge or even other structures are **typically** modeled with the use of **vertical** planes.

## Perpendicular Plane Properties

When two planes are **parallel** to one another, but at **right** angles to one another, we say that the planes are perpendicular to one another. That the angle formed by the 2 **planes** is exactly **ninety** degrees is **deduced** from this. **The** following is a list of some properties that perpendicular planes have:

- Only if the
**normal**vectors of the two**planes**are orthogonal can we say that the planes are indeed perpendicular to one another (i.e.,**perpendicular).**Only when one normal**vector’s projection**on the other is zero do the planes**become perpendicular.** - If the
**planes**are**perpendicular**to one another, then every line which lies in one plane and**therefore**is**perpendicular**to the normal vector of such a plane will also be**perpendicular**to the**corresponding**plane. If the planes are not perpendicular to one**another,**then the**planes**are parallel to one another. - If the
**planes**were**perpendicular**to one another, any line that really is parallel towards the**normal**vector**among**one plane will also be parallel to a normal**vector**of the other plane if the planes were**perpendicular**to one. - If two
**planes**are**parallel,**then the distance**between**two points is equivalent to the length of a vector first from the point to the**plane**that is**projected**over onto the**normal**vector of a plane.

## A Numerical Example of Perpendicular Planes

### Example

**Plane** 1 is **represented** by the normal vector** (2, 2, 3) while** the second **plane** is **represented **by a **normal** vector**(4, 4, 4). Show** whether the two planes are **perpendicular** or not.

### Solution

We **know that** if the dot **product** of the **given** vectors is not zero, the two **planes** are not **perpendicular.**

**The** dot **product** of two vectors **results** in 28 **which** is **not **equal to zero, so the given two planes are not perpendicular.

*All mathematical drawings and images were created withÂ GeoGebra.*