**13.1 Three-Dimensional Coordinate Systems**

In a plane, we use two coordinate axes, the x-axis and y-axis, that are perpendicular to each other to determine the location of any point in the plane.

A point in the plane can be represented as an **ordered pair** (*a*, *b*) of real numbers, where a is the x-coordinate and b is the y-coordinate.

Hence, a plane is called two-dimensional.

In space, we use three coordinate axes, the x-axis, y-axis and z-axis, (each of which is perpendicular to the other two) that meet at a point O, the origin.

A point in space can be represented as an **ordered triple** (a, b, c) of real numbers, where a is the x-coordinate, b is the y-coordinate and c is the z-coordinate.

Hence, space is called three-dimensional.

To locate a point P(*a*, *b*, *c*), we can start at the origin O, move a units along the x-axis, then b units parallel to the y-axis, and then c units parallel to the z-axis.

The three coordinate axes determine the three **coordinate planes.**

The **xy-plane** is the plane that contains the x- and y-axes;

the **yz-plane** is the plane that contains the y- and z-axes;

the **xz-plane** is the plane that contains the x- and z-axes.

If P(x, y, z) is a point in space, then

x is the directed distance from the yz-plane to the point P,

y is the directed distance from the xz-plane to the point P, and

z is the directed distance from the xy-plane to the point P.

For the point *P*(3, 2, 6), determine the directed distance from each coordinate plane.

The Cartesian product RxR is the set of all ordered pairs of real numbers and is denoted by R^{2}.

RxR = { (x, y) | x, y ε R }

The Cartesian product RxRxR is the set of all ordered triples of real numbers and is denoted by R^{3}.

RxRxR = { (x, y, z) | x, y, z ε R }

There is a one-to-one correspondence between the points in space and the ordered triples in R^{3}.

It is called a **three-dimensional rectangular coordinate system**.

The three coordinate axes divide space into eight regions called **octants**, and the first octant is the set of points whose coordinates are all positive.

In two-dimensional analytic geometry, the graph of an equation in the variables x and y is a **curve** in R^{2}.

In three-dimensional analytic geometry, the graph of an equation in the variables *x*, *y* and *z* is a **surface** in R^{3}.

Describe the surface in R^{3} represented by each equation.

(a) z = 3 (b) y = 5

Solution

(a). In R^{3} , the equation z = 3 represents the set of all points

{ (x, y, z) | x, y ε R, z = 3 }

This set contains infinitely many points.

However, each point (x, y, 3) in this set has a common property: each point lies 3 units above the xy-plane.

That is, this set of points is the **horizontal plane** that is parallel to the xy-plane and is 3 units above it.

(b) Similarly, in R^{3} , the equation y = 5 represents the set of all points

{ (x, y, z) | x, z ε R, y = 5 }

Each point (x, 5, z) in this set lies 5 units above the xz-plane.

So this set of points is the **vertical plane** that is parallel to the xz-plane and is 5 units to its right.

Let k be a constant (a real number). Then

(1) The graph of the equation x = *k* in R^{3} is a plane parallel to the yz-plane;

(2) The graph of the equation *y* = *k* in R^{3} is a plane parallel to the *xz*-plane;

(3) The graph of the equation *z* = *k* in R^{3} is a plane parallel to the *xy*-plane;

**YOU TRY IT**

Describe the region of R^{3} that is represented by the inequality y > 5.

The formula for the distance between two points in the plane can be extended to two points in space.

**Distance Formula in Three Dimensions**

The distance | P_{1}P_{2} | between the points P_{1}(x_{1}, y_{1}, z_{1}) and P_{2}(x_{2}, y_{2}, z_{2}) equals

Study the figure below: can you explain how to derive this formula? (Hint: use the Pythagorean theorem twice.)

(a) Find the distance between the points P(2, -1, 7) and Q(3, 1, 5).

(b) Find the equation of a sphere with radius r and center C(h, k, l).

Given three points in space, how are the distances between them related if the points lie on a straight line?

**Section 13.1 - Suggested Exercises**

# 1, 3, 7, 9, 11, 13, 15, 17, 21, 23, 25, 29, 33, 35