Section 13.1 Three-Dimensional Coordinate Systems

THE THREE-DIMENSIONAL COORDINATE SYSTEM z y x O There are three coordinate planes: xy-plane, xz- plane, and yz-plane. These three planes separate three-space into 8 octants.

COORDINATES IN THREE-SPACE The coordinates of a point P in three-space are (x, y, z) where x is its directed distance from the yz-plane; y is its directed distance from the xz-plane; z is its directed distance from the xy-plane.

PROJECTIONS The point P(a, b, c) determines a rectangular box with the origin. If we drop a perpendicular from P to the xy-plane, we get a point Q with coordinates (a, b, 0) called the projection of P on the xy-plane. Similarly, R(0, b, c) and S(a, 0, c) are the projections of P onto the yz-plane and xz-plane, respectively.

The Cartesian product is the set of all ordered triples of real numbers and is denoted by. We have given a one-to- one correspondence between points P in space and the ordered triples (a, b, c) in. It is called a three-dimensional rectangular coordinate system.

THE DISTANCE FORMULA The distance formula between two points P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) is given by

MIDPOINT FORMULA The coordinates of the midpoint of the line segment joining two point P 1 (x 1, y 1, z 1 ) and P 2 (x 2, y 2, z 2 ) are:

EQUATION OF A SPHERE An equation of a sphere with center C(h, k, l) and radius r is In particular, if the center is the origin O, then the equation of the sphere is x 2 + y 2 + z 2 = r 2