Three-Dimensional Space: Learn It 1

  • Describe three-dimensional space mathematically
  • Find and graph points in 3D space using their [latex](x, y, z)[/latex] coordinates
  • Write the distance formula in three dimensions
  • Write the equations for simple planes and spheres
  • Perform vector operations in ℝ3

Three-Dimensional Coordinate Systems

Vectors work well for solving two-dimensional problems, but real life happens in three dimensions. Whether you’re navigating mountainous terrain, analyzing the flight path of an aircraft, or studying molecular structures, you need tools that work in 3D space.

The two-dimensional coordinate system uses two perpendicular axes: the horizontal [latex]x[/latex]-axis and vertical [latex]y[/latex]-axis. To work in three dimensions, we add a third axis—the [latex]z[/latex]-axis—perpendicular to both existing axes.

three-dimensional rectangular coordinate system

Consists of three mutually perpendicular axes: the [latex]x[/latex]-axis, [latex]y[/latex]-axis, and [latex]z[/latex]-axis. Since each axis represents all real numbers, we denote this system as [latex]\mathbb{R}^3[/latex].

The Right-Hand Rule

In Figure 1(a), the positive [latex]z[/latex]-axis is shown above the plane containing the [latex]x[/latex]– and [latex]y[/latex]-axes. The positive [latex]x[/latex]-axis appears to the left and the positive [latex]y[/latex]-axis is to the right. A natural question to ask is: How was arrangement determined?

The arrangement of the three axes follows a standard convention called the right-hand rule. Here’s how it works:

  1. Point your right hand’s fingers along the positive [latex]x[/latex]-axis
  2. Curl your fingers toward the positive [latex]y[/latex]-axis
  3. Your thumb naturally points in the direction of the positive [latex]z[/latex]-axis

This standard ensures consistency across mathematical applications. While left-hand rule systems exist, we always use the right-hand rule convention.

This figure has two images. The first is a 3-dimensional coordinate system. The x-axis is forward, the y-axis is horizontal to the left and right, and the z-axis is vertical. The second image is the 3-dimensional coordinate system axes with a right hand. The thumb is pointing towards positive z-axis, with the fingers wrapping around the z-axis from the positive x-axis to the positive y-axis.
Figure 1. (a) We can extend the two-dimensional rectangular coordinate system by adding a third axis, the [latex]z[/latex]-axis, that is perpendicular to both the [latex]x[/latex]-axis and the [latex]y[/latex]-axis. (b) The right-hand rule is used to determine the placement of the coordinate axes in the standard Cartesian plane.

Plotting Points in 3D

In two dimensions, we describe a point in the plane with the coordinates [latex](x, y)[/latex]. In three dimensions, you add a third coordinate, [latex]z[/latex]: [latex](x, y, z)[/latex]. A point in space is identified by all three coordinates (Figure 2).

To plot the point [latex](x, y, z)[/latex]:

  1. Move [latex]x[/latex] units along the [latex]x[/latex]-axis
  2. Move [latex]y[/latex] units in the direction of the [latex]y[/latex]-axis
  3. Move [latex]z[/latex] units in the direction of the [latex]z[/latex]-axis

The intersection of these three movements locates your point in 3D space.

This figure is the positive octant of the 3-dimensional coordinate system. In the first octant there is a rectangular solid drawn with broken lines. One corner is labeled (x, y, z). The height of the box is labeled “z units,” the width is labeled “x units” and the length is labeled “y units.”
Figure 2. To plot the point [latex](x,y,z) [/latex] go [latex] x [/latex] units along the [latex]x[/latex]-axis, then [latex]y[/latex] units in the direction of the [latex]y[/latex]-axis, then [latex] z[/latex] units in the direction of the z-axis.
 Three-dimensional plotting can be challenging to visualize on paper. Try using physical objects or online 3D graphing tools to better understand how points relate to the three axes
Sketch the point [latex](1, -2, 3)[/latex] in three-dimensional space.

Coordinate Planes

Just as two perpendicular axes define a coordinate plane in 2D, pairs of axes define coordinate planes in 3D. With three axes, we get three coordinate planes:

  • [latex]xy[/latex]-plane: Contains the [latex]x[/latex] and [latex]y[/latex] axes, where [latex]z = 0[/latex]
  • [latex]xz[/latex]-plane: Contains the [latex]x[/latex] and [latex]z[/latex] axes, where [latex]y = 0[/latex]
  • [latex]yz[/latex]-plane: Contains the [latex]y[/latex] and [latex]z[/latex] axes, where [latex]x = 0[/latex]

Formally, these planes are defined as:

  • [latex]xy[/latex]-plane: [latex]{(x,y,0) : x,y \in \mathbb{R}}[/latex]
  • [latex]xz[/latex]-plane: [latex]{(x,0,z) : x,z \in \mathbb{R}}[/latex]
  • [latex]yz[/latex]-plane: [latex]{(0,y,z) : y,z \in \mathbb{R}}[/latex]
To visualize this, imagine standing in a corner room with two finished walls meeting at a right angle. If you face outward from the corner: the floor is the [latex]xy[/latex]-plane, the wall to your right is the [latex]xz[/latex]-plane, and the wall to your left is the [latex]yz[/latex]-plane.
This figure is the first octant of a 3-dimensional coordinate system. Also, there are the x y-plane represented with a rectangle with the x and y axes on the plane. There is also the x z-plane on the x and z axes and the y z-plane on the y and z axes.
Figure 5. The plane containing the [latex]x[/latex]– and [latex]y [/latex]-axes is called the [latex]xy [/latex]-plane. The plane containing the [latex]x [/latex]– and [latex]z [/latex]-axes is called the [latex]xz [/latex]-plane, and the [latex]y [/latex]– and [latex]z [/latex]-axes define the [latex]yz [/latex]-plane.

In 2D, the coordinate axes divide the plane into four quadrants. In 3D, the three coordinate planes divide space into eight regions called octants.

This figure is the 3-dimensional coordinate system with the first octant labeled with a roman numeral I, I, II, III, IV, V, VI, VII, and VIII. Also, for each quadrant there are the signs of the values of x, y, and z. They are: I (+, +, +); 2nd (-, +, +); 3rd (-, -, +); 4th (+, -, +); 5th (+, +, -); 6th (-, +, -); 7th (-, -, -); and 8th (+, -, -).
Figure 6. Points that lie in octants have three nonzero coordinates.

Distance Formula in Three Dimensions

The distance formula extends naturally from 2D to 3D. In two dimensions, the distance between points [latex](x_1, y_1)[/latex] and [latex](x_2, y_2)[/latex] is:

[latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/latex]

The distance [latex]d[/latex] between points [latex](x_1, y_1, z_1)[/latex] and [latex](x_2, y_2, z_2)[/latex] is:

[latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[/latex]

distance formula in 3D

The distance [latex]d[/latex] between points [latex](x_1, y_1, z_1)[/latex] and [latex](x_2, y_2, z_2)[/latex] is given by the formula:

[latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}[/latex].

This formula represents the length of the diagonal of a rectangular prism (box) with the two points as opposite corners.

This figure is a rectangular prism. The lower, left back corner is labeled “P sub 1=(x sub 1,y sub 1,z sub 1). The lower front right corner is labeled “(x sub 2, y sub 2, z sub 1)”. There is a line between P sub 1 and P sub 2 and is labeled “d sub 1”. The upper front right corner is labeled “P sub 2=(x sub 2,y sub 2,z sub 2).” There is a line from P sub 1 to P sub 2 and is labeled “d (P sub 1,P sub 2).” The front right vertical side is labeled “|z sub 2-z sub 1|”.
Figure 7. The distance between [latex]P_1[/latex] and [latex]P_2[/latex] is the length of the diagonal of the rectangular prism having [latex]P_1[/latex] and [latex]P_2[/latex] as opposite corners.
Find the distance between points [latex]P_1=(3, -1, 5)[/latex] and [latex]P_2=(2, 1, -1)[/latex].

This figure is the 3-dimensional coordinate system. There are two points. The first is labeled “P sub 1(3, -1, 5)” and the second is labeled “P sub 2(2, 1, -1)”. There is a line segment between the two points.
Figure 8. Find the distance between the two points.

Key Differences in Three Dimensions

Before moving on to the next section, let’s get a feel for how [latex]\mathbb{R}^3[/latex] differs from [latex]\mathbb{R}^2[/latex].

Three-dimensional space offers more geometric flexibility than 2D:

In 2D: Non-parallel lines must intersect.

In 3D: Non-parallel lines can be skew lines—they don’t intersect and aren’t parallel.

This figure is the 3-dimensional coordinate system. There is a line drawn at z = 3. It is parallel to the x y-plane. There is also a line drawn at y = 2. It is parallel to the x-axis.
Figure 9. These two lines are not parallel, but still do not intersect.

 

In 2D: Circles either intersect, are tangent, or are completely separate.

In 3D: Circles can be interconnected (like links in a chain) without sharing any points.

This figure is the 3-dimensional coordinate system. There are two circles drawn. The first circle is centered around the z-axis, at z = 1. The second circle has the positive x-axis as its diameter. It intersects the x-axis at x = 0 and x = 6. It is vertical.
Figure 10. These circles are interconnected, but have no points in common.

This increased flexibility makes 3D geometry richer and more complex than 2D geometry.