Three-Dimensional Coordinate Systems

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Section 13.1

• Three-Dimensional Coordinate Systems

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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.
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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.
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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.
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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.
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THE DISTANCE FORMULA
The distance formula between two points
P1(x1, y1, z1) and P2(x2, y2, z2) is given by
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MIDPOINT FORMULA
The coordinates of the midpoint of the line
segment joining two point P1(x1, y1, z1) and
P2(x2, y2, z2) are
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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