Download 10.1 Three-Dimensional Coordinate Systems

Arkansas Tech University
MATH 2934: Calculus III
Dr. Marcel B. Finan
10.1
Three-Dimensional Coordinate Systems
In this section, we learn the aspects of the three-dimensional coordinate
system. We define the three dimensional coordinate system as shown
in Figure 10.1.1(a).
Figure 10.1.1
There are three coordinate axes: The x−axis, the y−axis, and the z−axis.
They are oriented as shown in Figure 10.1.1(a). The point O is called the
origin of the system. As shown in Figure 10.1.1(b), the coordinate system
determine three coordinate planes and these planes divide the space into
eight parts, each called an octant: Four octants above the xy−plane and
four octants below the xy−plane. The octants are numbered as shown in
Figure 10.1.2.
Figure 10.1.2
The first octant is determined by the positive x−, y−, and z−axes.
In the two-dimensional system two numbers are required to determine a
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point. Likewise, in the three-dimensional coordinate system, a point in
space P requires three numbers: a, b, and c. More precisely, a point P in
space is uniquely determined by an ordered triple (a, b, c) as shown in Figure
10.1.3.
Figure 10.1.3
The numbers 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. In terms of sets, the
three-dimensional coordinate system will be denoted by the set
IR3 = {(x, y, z) : x, y, z ∈ IR}.
Now, from the point P (a, b, c) we can create a box with each vertex having
the coordinates shown in Figure 10.1.4. The point Q(a, b, 0) is called the
orthogonal projection of P onto the xy−plane. Likewise, R(0, b, c) is the
orthogonal projection of P onto the zy−plane and S(a, 0, c) is the orthogonal
projection of P onto the xz−plane.
Figure 10.1.4
The solutions to an equation of the form f (x, y, z) = 0 are represented by a
surface in IR3 .
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Example 10.1.1
Sketch and describe each of the following surfaces in IR3 : (a) y = 3 (b)
y = x.
Solution.
(a) The surface is a plane parallel to the xz− plane and located 3 units to
the right of it as shown in Figure 10.1.5(a).
(b) The surface is a plane that contains the z−axis, and intersects the
xy−plane at the line y = x as shown in Figure 10.1.5(b)
Figure 10.1.5
The Distance Formula
The distance between two points P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) is given by
the distance formula
p
||P1 P2 || = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
This formula is easily derived from Figure 10.1.6. Indeed, using the Pythagorean
formula in the highlighted right triangle, we find
||P1 P2 ||2 = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
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Figure 10.1.6
Example 10.1.2
Find the distance between the two points C(a, b, c) and P (x, y, z).
Solution.
p
Using the distance formula, we find ||CP || = (x − a)2 + (y − b)2 + (z − c)2
Example 10.1.3
Describe and sketech the surface (x − a)2 + (y − b)2 + (z − c)2 = r2 , where
r > 0.
Solution.
The collection of all points in space whose distances from a fixed point
C(a, b, c) is equal to a positive number r is called a sphere. We call the
point C the center and the number r the radius of the sphere. This
definition, implies that ||CP || = r or equivalently
(x − a)2 + (y − b)2 + (z − c)2 = r2 .
This last equation is called the standard form of the equation of a sphere.
The surface is shown in Figure 10.1.7
Figure 10.1.7
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Example 10.1.4
Find the center and the radius of the sphere: 2x2 +2y 2 +2z 2 +6x+4y−2z = 1.
Solution.
Using the method of completing the square, we find
2x2 + 2y 2 + 2z 2 + 6x + 4y − 2z =1
1
9
9
1
2
2
2
+ 2(y + 2y + 1) + 2 z − z +
=1 + + 2 +
2 x + 3x +
4
4
2
2
2
2
1
3
+ 2(y + 1)2 + 2 z −
=8
2 x+
2
2
3 2
1 2
2
x+
+ (y + 1) + z −
=4.
2
2
Hence, the center is C − 23 , −1, 12 and the radius is 2
Example 10.1.5
Describe the region in IR3 represented by the compound inequality 1 ≤
x2 + y 2 + z 2 ≤ 4, z ≤ 0.
Solution.
p
An equivalent form of the given inequality is 1 ≤ x2 + y 2 + z 2 ≤ 2. This
inequality represents the points in IR3 whose distance from the origin is at
least 1 and at most 2. Since z ≤ 0, we consider only those points that lie on
or below the xy−plane. Thus, the inequality represents those points that
are between (or on) the sphere x2 + y 2 + z 2 = 1 and x2 + y 2 + z 2 = 4 and
beneath (or on) the xy−plane as shown in Figure 10.1.8
Figure 10.1.8
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