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Section 13.1

Section 13.1. Three-Dimensional Coordinate Systems. z. y. O. x. THE THREE-DIMENSIONAL COORDINATE SYSTEM. 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.

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Section 13.1

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  1. Section 13.1 Three-Dimensional Coordinate Systems

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

  3. COORDINATES INTHREE-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.

  4. 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.

  5. 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.

  6. THE DISTANCE FORMULA The distance formula between two points P1(x1,y1,z1) and P2(x2,y2,z2) is given by

  7. MIDPOINT FORMULA The coordinates of the midpoint of the line segment joining two point P1(x1,y1,z1) and P2(x2,y2,z2) are:

  8. 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 x2 + y2 + z2 = r2

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