In 3D space (also called *xyz* space), the **xy plane** contains the x-axis and y-axis:

The xy plane can be described as **the set of all points (x, y, z) where z = 0. ** In other words, any point (x, y, 0). For example, all of the following points are on the xy plane:

- (1, 5, 0)
- (-2, 19, 0)
- (π, -1, 0)
- (.5, .2, 0)

This fact gives us **the equation for the xy plane:** z = 0.

This is just an extension of the same idea of the x-axis (in the Cartesian plane) being the place where y = 0:

The xy plane, together with the yz plane and xz plane, divide space into eight *octants*. The *O* in the center of the diagram is the origin, which is a starting point for the 3D-coordinate system. The points are described by an *ordered triple* of real numbers (x, y, z). For example, the point (2, 3, 0) can be found at:

- x = 2,
- y = 3,
- z = 0.

As z is zero, we know this point must be somewhere on the xy plane.

## Distance Formula for Points in the XY Plane

The distance between any two points in xyz-space can be found with a generalization of the distance formula:

**Example question**: What is the distance between the points (4, 3, 0) and (2, 9, 0)?

Step 1: **Identify the coordinate components **that we need to put into the formula. We know our coordinates are always ordered (x, y, z), so:

- (4, 3, 0):
- x
_{1}= 4 - y
_{1}= 3 - z
_{1}= 0.

- x
- (2, 9, 0):
- x
_{2}= 2 - y
_{2}= 9 - z
_{z}= 0.

- x

Don’t worry about which coordinate is which (e.g. does x = 4 go into x_{1} or x_{2}?). The distance formula squares these values, so you’ll get the same answer no matter which way you choose.

Step 2: **Plug your values from Step 1 into the distance formula**:

If you aren’t good with algebra, head over to Symbolab and just replace the x, y, z values with your inputs.

**CITE THIS AS:**

**Stephanie Glen**. "xy Plane" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/xy-plane/

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