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Three Dimensional Coordinates

Three Dimensional Coordinates

A system for locating points in three dimensions. This is typically done using (x, y, z) coordinates, but any variables may be used.

Example: The point (1, 2, 3) plotted in (x, y, z) coordinates

3D coordinate system with x, y, z axes showing point (1, 2, 3) marked with dashed red lines indicating its position.

 

See also

Key Formula

P=(x, y, z)P = (x,\, y,\, z)
Where:
  • xx = The signed distance from the origin along the x-axis (left–right)
  • yy = The signed distance from the origin along the y-axis (forward–backward)
  • zz = The signed distance from the origin along the z-axis (up–down)

Worked Example

Problem: Find the distance between the points A = (1, 2, 3) and B = (4, 6, 3) in three-dimensional space.
Step 1: Write the 3D distance formula, which extends the Pythagorean theorem to three dimensions.
d=(x2−x1)2+(y2−y1)2+(z2−z1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
Step 2: Substitute the coordinates of A and B into the formula.
d=(4−1)2+(6−2)2+(3−3)2d = \sqrt{(4-1)^2 + (6-2)^2 + (3-3)^2}
Step 3: Compute each squared difference.
d=32+42+02=9+16+0d = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16 + 0}
Step 4: Add the values and take the square root.
d=25=5d = \sqrt{25} = 5
Answer: The distance between A and B is 5 units.

Another Example

Problem: Plot and describe the location of the point P = (−2, 3, 5) in three-dimensional space.
Step 1: Start at the origin (0, 0, 0), where all three axes intersect.
Step 2: Move 2 units in the negative x-direction (to the left along the x-axis).
x=−2x = -2
Step 3: From that position, move 3 units in the positive y-direction.
y=3y = 3
Step 4: Finally, move 5 units upward in the positive z-direction. You have arrived at P.
P=(−2, 3, 5)P = (-2,\, 3,\, 5)
Answer: The point P is located 2 units left, 3 units forward, and 5 units up from the origin.

Frequently Asked Questions

What is the difference between 2D and 3D coordinates?
Two-dimensional coordinates use two values (x,y)(x, y) to locate a point on a flat plane. Three-dimensional coordinates add a third value zz, which represents height or depth, allowing you to pinpoint a location in space rather than just on a flat surface.
How do you plot a point in 3D coordinates?
Start at the origin where the three axes meet. Move along the x-axis by the first coordinate, then parallel to the y-axis by the second coordinate, and finally parallel to the z-axis by the third coordinate. The spot you reach is the plotted point.

2D Coordinates (x, y) vs. 3D Coordinates (x, y, z)

Two-dimensional coordinates describe positions on a flat plane using two perpendicular axes. Three-dimensional coordinates extend this system by adding a third perpendicular axis, enabling you to describe positions in space. Every 2D point can be thought of as a 3D point with z=0z = 0. In 2D the distance formula uses two squared terms; in 3D it uses three.

Why It Matters

Three-dimensional coordinates are essential for describing the physical world, since real objects occupy space with width, depth, and height. They form the backbone of fields like physics (tracking motion of particles), engineering (designing structures), computer graphics (rendering 3D models and video games), and navigation (GPS uses latitude, longitude, and altitude—three coordinates). Without a 3D coordinate system, representing and computing spatial relationships mathematically would not be possible.

Common Mistakes

Mistake: Mixing up the order of coordinates, such as writing (z,y,x)(z, y, x) instead of (x,y,z)(x, y, z).
Correction: The order always matters. By convention, the first value is xx, the second is yy, and the third is zz. Swapping them places the point in a completely different location.
Mistake: Forgetting the z-component when computing distance, effectively using the 2D distance formula in a 3D problem.
Correction: Always include all three squared differences in the distance formula: (x2−x1)2+(y2−y1)2+(z2−z1)2(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2. Even if one difference is zero, include it to avoid errors.

Related Terms

  • Point — A location described by coordinates
  • Coordinates — General system for specifying positions
  • Coordinate Plane — The 2D version of the coordinate system
  • Number Line — The 1D version of the coordinate system
  • Variable — Letters used to represent each coordinate
  • Distance Formula — Calculates distance between two points
  • Origin — The reference point (0, 0, 0) in 3D