**Introduction to Rectangular 3D Coordinates**

Figure 13-2 illustrates the simplest possible set of *rectangular 3D coordinates,* also called *Cartesian three-space* or *xyz-space* . All three number lines have equal increments. (This is a perspective illustration, so the increments on the z axis appear distorted. A true 3D rendition would have the positive z axis perpendicular to the page.) The three number lines all intersect at a single point, the origin, which corresponds to the zero points on each line.

**Fig** . 13-2. Cartesian three-space, also called xyz-space.

The horizontal (right-and-left) axis is called the *x* axis; the vertical (up-and-down) axis is called the *y* axis, and the page-perpendicular (in-and-out) axis is called the z axis. In most renditions of rectangular 3D coordinates, the positive *x* axis runs from the origin toward the viewer’s right, and the negative *x* axis runs toward the left. The positive *y* axis runs upward, and the negative *y* axis runs downward. The positive z axis “points out of the page directly at you,” and the negative z axis “points behind the page straight away from you.” Sometimes you will see graphs in which the *y* and z axes are interchanged from the way they’re shown here, so the *y* axis is perpendicular to the page while the z axis runs up and down.

**Ordered Triples as Points**

Figure 13-3 shows two specific points, called P and Q, plotted in Cartesian three-space. The coordinates of point P are (−5,−4,3), and the coordinates of point Q are (3,5,−2). Points are denoted as *ordered triples* in the form (x, y, z), where the first number represents the value on the *x* axis, the second number represents the value on the *y* axis, and the third number represents the value on the *z* axis. The word “ordered” means that the order, or sequence, in which the numbers are listed is important. The ordered triple (1,2,3) is not the same as any of the ordered triples (1,3,2), (2,1,3), (2,3,1), (3,1,2), or (3,2,1), even though all of the triples contain the same three numbers.

**Fig. 13-3.** Two points in Cartesian three-space.

In an ordered triple, there are no spaces after the commas, as there are in the notation of a set or sequence.

**Variables, Origin, and Distance Between Points**

**Variables and Origin**

In Cartesian three-space, there are usually two independent-variable coordinate axes and one dependent-variable axis. The *x* and *y* axes represent independent variables; the *z* axis represents a dependent variable whose value is affected by both the *x* and the *y* values.

In some scenarios, two of the variables are dependent and only one is independent. Most often, the independent variable in such cases is *x* . Rarely, you’ll come across a situation in which none of the values depends on either of the other two, or when a correlation, but not a true relation, exists among the values of two or all three of the variables. Plots of this sort usually look like “swarms of points,” representing the results of observations, or values predicted by some scientific theory. These graphs are known as *scatter plots,* and are common in statistics.

**Distance Between Points**

Suppose there are two different points P = ( *x* _{0} ,y _{0} ,z _{0} ) and Q = ( *x* _{1} ,y _{1} ,z _{1} )in Cartesian three-space. The distance *d* between these two points can be found using this formula:

d = [(x _{1} - *x* _{o} ) ^{2} + (y _{1} - *y* _{o} ) ^{2} + (z _{1} - z _{o} ) ^{2} ] ^{½}

**Rectangular 3D Coordinates Practice Problems**

**Practice 1**

What is the distance between the points P = (−5,−4,3) and Q = (3,5,−2) illustrated in Fig. 13-3? Express the answer rounded off to three decimal places.

**Solution 1**

We can plug the coordinate values into the distance equation, where:

x _{0} =−5

*x* _{1} = 3

y _{0} =−4

*y* _{1} = 5

*z _{o} = 3*

*z* _{1} *=* −2

Therefore:

d = {[3 −(−5)] ^{2} + [5 −(−4)] ^{2} + (−2 − 3) ^{2} } ^{½}

= [8 ^{2} + 9 ^{2} + (−5) ^{2} ] ^{½}

= (64 + 81 + 25) ^{½}

= 170 ^{½}

= 13.038

Find practice problems and solutions for these concepts at the Vectors and 3D Practice Test.

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