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):
- x1 = 4
- y1 = 3
- z1 = 0.
- (2, 9, 0):
- x2 = 2
- y2 = 9
- zz = 0.
Don’t worry about which coordinate is which (e.g. does x = 4 go into x1 or x2?). 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.
Stephanie Glen. "xy Plane" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/xy-plane/
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