**Introduction to Rectangular 3D Coordinates**

Figure 9-3 illustrates the simplest possible set of *rectangular 3D coordinates* . 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 intersect at their zero points.

**Each division equals 1 unit**

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 comes “out of the page,” and the negative *z* axis extends “back behind the page.”

**Ordered Triples As Points**

Figure 9-4 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.

In an ordered triple, there are no spaces after the commas, as there are in the notation of a set or sequence. The rule is the same as that for ordered pairs.

**Variables, Origin, and the 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 mathematical 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.

**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* _{0} ) ^{2} + ( *y* _{1} − *y* _{0} ) ^{2} + ( *z* _{1} − *z* _{0} ) ^{2} ] ^{1/2}

**Three Number Lines Practice Problem**

**PROBLEM 1**

What is the distance between the points *P* = (−5, −4, 3) and *Q* = (3, 5, −2) illustrated in Fig. 9-4? 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* _{0} = 3

*z* _{1} = −2

Therefore:

*d* = {[3 − (−5)] ^{2} + [5 − (−4)] ^{2} + (−2 −3) ^{2} } ^{1/2}

= [8 ^{2} + 9 ^{2} + (−5) ^{2} ] ^{1/2}

= (64 + 81 + 25) ^{1/2}

= 170 ^{1/2}

= 13.038

Practice problems for these concepts can be found at: Vectors And Cartesian Three-Space Practice Test.

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