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Section 11.1 – Three-Dimensional Coordinate
Systems
Any point in a plane can be represented by an ordered pair (a, b) of real numbers, where
a is the x-coordinate and b is the y-coordinate. A point in three dimensional space is
represented by an ordered triple (a, b, c) of real numbers.
Definitions.
1. To locate a point in three-dimensional space we first choose a fixed point O (the
origin) and three directed lines through O that are perpendicular to each other,
called the coordinate axes and labeled the x-axis, y-axis, and z-axis. Then each
point in the three-dimensional space can be represented by an ordered triple
(a, b, c) where a, b, and c are the x-, y- and z-coordinates respectively.
2. The right-hand rule determines the direction of the z-axis. If you curl the
fingers of your right hand around the z-axis in the direction from the positve
x-axis to the positive y-axis, then your thumb points in the positive direction of
the z-axis.
3. The three coordinate axes determine the three coordinate planes shown below.
The xy-plane is the plane that contains the x- and y-axes. Similarly, the xz-plane
is the plane that contains the x- and z-axes, and the yz-plane is the plane that
contains the y- and z-axes. The coordinate planes divide space into eight parts,
called octants. The first octant is the octant where all the coordinate axes are
positive.
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4. Let P be any point in space with a the directed distance from P to the yz-plane,
b the directed distance from P to the xz-plane, and c the directed distance from
P to the xy-plane. Then we represent P by the ordered triple (a, b, c) and a, b,
and c are the coordinates of P ; a is the x-coordinate, b is the y-coordinate, and
c is the z-coordinate. The first octant is the point where all the coordinates are
positive.
5. The point P determines a rectangular box as shown above. If we drop a perpendicular line from the point P (a, b, c) to the xy-plane, we get a point Q with
coordinates (a, b, 0) called the projection of P on the xy-plane. Similarly, the
points R(0, b, c) and S(a, 0, c) are the projections of P on the yz-plane and xzplane, respectively.
6. The Cartesian product
R × R × R = {(x, y, z) | x, y, z ∈ R}
is the set of all ordered triples of real numbers and is denoted R3 . We have given
a one-to-one correspondence between points in space and ordered triples in R3 .
This is called a three-dimensional rectangular coordinate system.
(Note that R represents the real numbers which correspond to a one-dimensional
line, and R2 represents ordered pairs of real numbers which correspond to the
normal two-dimension plane. )
Example 1. Plot the points (−4, 3, 5) and (3, −2, −6).
Example 2. What is the projection of the point (−4, 3, 5) onto the xy-plane, the
xz-plane, and the yz-plane?
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In two-dimensional geometry, the graph of an equation involving x and y is a curve in
R2 . In three-dimensional geometry, an equation in x, y, and z represents a surface in
R3 .
Example 3. What surfaces in R3 are represented by the following equations?
(a) z = 5
(b) x = −3
Note. The context of an equation will determine whether the equation represents a
curve in R2 or a surface in R3 . For comparison graph x = −3 in R2 .
Example 4. Describe and sketch the surface in R3 represented by the equation y = x.
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Distance Formula in Three Dimensions. The distance |P1 P2 | between the points
P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) is
p
|P1 P2 | = (x1 − x2 )2 + (y1 − y2 )2 + (z1 − z2 )2
Example 5. Find the lengths of the sides of the triangle ABC with A(3, −4, 1),
B(5, −3, 0), and C(6, −7, 4). Determine if the triangle is isosceles, a right triangle,
both, or neither.
Example 6. Find an equation of the sphere with radius r and center C(h, k, l).
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Equation of a Sphere. An equation of a sphere with center C(h, k, l) and radius r is
(x − h)2 + (y − k)2 + (z − l)2 = r2 .
In particular, if the center is the origin O, then an equation of a sphere is
x2 + y 2 + z 2 = r2 .
Example 7. Find the equation of a sphere if one of its diameters has endpoints
(3, −4, 1) and (5, −3, 0). (Note we calculated this distance in Example 5.)
Example 8. What is the intersection of the sphere from Example 7 with the xy-plane?
Example 9. Find the equation of a sphere with center (−1, 4, 0) if it touches the
xz-plane.
Example 10. Write the equation of the sphere given by x2 − 4x + y 2 + 2y + z 2 − 3z = 4
in standard form. Find the center and radius.
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Example 11. Describe the regions of R3 represented by the following inequalities.
(a) z < 1
(b) x2 + y 2 = 4
(c) x2 + 2x + y 2 + z 2 − 4z ≥ 4