Contents
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Euclidean Geometry in 3-Dimensional Space
1.1 Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1
Review: Cartesian Coordinates in the Plane . . . . . . .
1.1.2
Cartesian Coordinates in 3-space . . . . . . . . . . . . .
1.1.3
Using the 3-Dimensional Cartesian Coordinate System
1.1.2
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.3
Solutions to Exercises . . . . . . . . . . . . . . . . . . . .
Calculus III
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
CONTENTS
Calculus III
Last update: August 18, 2015
CONTENTS
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
Chapter 1
Euclidean Geometry in 3-Dimensional Space
Comments and suggestions are welcome!
This chapter is devoted to developing the background and tools needed to study differential
and integral calculus in 3-dimensional space. The ideas discussed extend far beyond 3
dimensions, however, and can be applied to the study of n-dimensional space for any
positive integer n.
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
Comments and suggestions are welcome!
1.1
Cartesian Coordinates
1.1.1
Review: Cartesian Coordinates in the Plane
The 2-dimensional or x y Cartesian coordinate plane is formed
by arranging two coordinate lines at right angles, with the positive direction of these lines to the right and up as shown in Figure
1 to the left. The horizontal axis is the x-axis and the vertical
axis is the y-axis. Such a coordinate system has an associated
coordinate grid constructed using lines parallel to the x- and
y-coordinate axes. Each point in the plane lies at the intersection
of a vertical line x = a and a horizontal line y = b.
The utility of this arrangement of axes is that we can use the
coordinate grid to assign a unique ordered pair of real numbers
(a, b) to each point in the plane. The first number a in the pair
is the x-coordinate of the point, and the second number b in
the pair is the y-coordinate of the point. In Figure 2 at the left
the coordinate pairs (a, b) for 2 different points are shown. The
point P in the first quadrant lies at the intersection of the vertical
line x = 1.5 and the horizontal line y = 3, and thus P is assigned
the Cartesian coordinates (1.5, 3). We use the notation P(1.5, 3)
to denote this fact. Similarly the point Q has Cartesian coordinates Q(−3, −2). It is important to note that the assignment
of coordinates is unique. That is, each point P in the plane is
assigned one and only one ordered pair of real numbers (a, b),
and each ordered pair of real numbers (a, b) corresponds to one
and only one point P in the plane.
Figure 1
Figure 2
● Distance from P to Q
Because the x- and y-axes intersect at a right angle we can use
the Theorem of Pythagoras to express the distance between two
points P and Q in the plane in terms of their coordinates. Figure 3
at the left shows the right triangle formed by the line connecting
P and Q, the vertical line through P and the horizontal line
through Q. The bottom leg of the triangle has length
1.5 − (−3) = 4.5 units
and the vertical right leg of the triangle has length
3 − (−2) = 5 units
Figure 3
Calculus III
Last update: August 18, 2015
The distance from P to Q, which of course equals the distance
from Q to P, is the length of the hypotenuse and is given, using
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
the Pythagorean Theorem, by
√
√
4.52 + 52 = 45.25 units
More generally, given two points P(x1 , y1 ) and Q(x2 , y2 ) in the
plane, the distance d(P, Q) from P to Q is given by the Euclidean distance formula:
d(P, Q) =
√
(x1 − x2 )2 + (y1 − y2 )2
(1)
This formula is the basis for Euclidean geometry in the plane.
● Equation of a Circle
The geometric definition of the circle with center (h, k) and
radius R is the set of all points at distance exactly R units from
the point with coordinates (h, k), as shown in Figure 4 at the
left. Letting (x, y) denote the coordinates of an arbitrary point
on this circle, we can use the Euclidean distance formula (1) to
write the equation of this circle, namely
√
(2)
R = (x − h)2 + (y − k)2
Squaring both sides and rearranging leads to the more familiar
form for the equation of a circle with center at (h, k) and radius
R:
Figure 4
(x − h)2 + (y − k)2 = R 2
1.1.2
Last update: August 18, 2015
Cartesian Coordinates in 3-space
By adding a third coordinate line, the z-axis, perpendicular to
the x y Cartesian plane, we can construct a Cartesian coordinate
system in 3-dimensional space, also referred to as 3-space. Envision looking down at a 2-dimensional Cartesian coordinate
system on a desk. To construct the 3-dimensional Cartesian coordinate system we place the z-axis perpendicular to both the
x− and y-axes, i.e., it would pass perpendicularly through the
desk. This leaves us with the question of which direction we
should make the positive side of the z-axis, above or below. The
standard choice is the right-hand orientation with the positive
side of the z-axis above the desk. The reason it is referred to as
the right-hand system as shown in Figure 5 at the left.
Figure 5
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
• The x− and y-axes are two perpendicular lines that define
the x y-coordinate plane.
• The x− and z-axes are two perpendicular lines that define
the xz-coordinate plane.
• The y− and z−axes are two perpendicular lines that define
the yz-coordinate plane.
In Figure 6 we have such a 3-dimensional Cartesian coordinate system in standard perspective in which the viewer is looking
down from above the x y-plane: the positive x-axis is going off to
the left, the positive y-axis points to the right, the positive z-axis
points upward, and the axes are mutually perpendicular. Just like
the 2-dimensional Cartesian coordinate plane was divided into
quadrants, the 3 coordinate planes defined by these axes divide
3-dimensional space into 8 subregions called octants. The octant
where the 3 coordinates are all positive is called the first octant.
The words “first octant” will occur in many problems that
you work this semester because of the ease in visualizing and
describing regions and objects in the first octant. The other
octants are not numbered.
Figure 6
1.1.3
Using the 3-Dimensional Cartesian Coordinate System
● Coordinates of a Point
To each point P in 3-dimensional space one assigns an ordered
triple of real numbers (x, y, z) that are the coordinates of that
point. In Figure 7 the coordinates (x1 , y1 , z1 ) are assigned to the
point P that sits in the first octant. One first drops a perpendicular line to the point Q in the x y-plane, with the distance from the
point to the plane giving the z-coordinate z = z1 of the point P.
Then, by constructing perpendicular lines to the x- and y-axes
from the point Q, we determine the x- and y-coordinates x = x1
and y = y1 exactly as we would in 2-dimensional space.
Figure 7
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CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
● Distance From P to Q
To find the formula for the distance from point P = (x1 , y1 , z1 ) to point Q = (x2 , y2 , z2 ) we first
project the points into the x y-plane as shown in Figure 8. Using the coordinates (x1 , y1 ) and
(x
√ 2 , y2 ) to form the right triangle ABC, we then use the Pythagorean theorem to find the length
(x2 − x1 )2 + (y2 − y1 )2 of the blue hypotenus AC in the figure.
Figure 8
The blue line AC in turn determines the blue line PR in the upper right triangle in Figure 9.
Figure 9
Using the Pythagorean Theorem a second time on the triangle PQR we find the length of the red
hypotenuse in the upper right triangle, which is the distance from P to Q, to be
√
Distance from P to Q = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2
(4)
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CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
● Equations of Basic Planes
The fact that the coordinate axes meet at right angles allows us
to easily write the equation of any plane that is parallel to one of
the 3 coordinate planes. For example, the x y-plane is the set of
all points whose coordinates (x, y, z) satisfy the equation
z=0
(5)
Notice that this single equation places no restrictions on the x
and y coordinates. This means that any point with coordinates
(x, y, 0) will lie on the x y-plane.
Next consider an equation for the plane that passes through
the point (1, 2, 3) and is parallel to the xy-plane. Being parallel
to the x y-plane means that the x and y coordinates of points
on this plane must again be unrestricted, but the z-coordinate
must always be z = 3. Hence an equation for the plane passing
through the point (1, 2, 3) parallel to the x y-plane is given by
Figure 10
z=3
(6)
In Figure 10 we have three planes given by the equations z = −4,
z = 3 and z = 7, all parallel to the (blue) z = 0 plane.
We can also study at the set of points whose coordinates
satisfy an inequality such as
z>3
Using the standard perspective, these are the points above the
plane given by the equation z = 3.
Any plane that is parallel to the yz-plane has an equation of
the form
x = constant
(7)
Figure 11
and any plane that is parallel to the xz-plane has an equation of
the form
y = constant
(8)
In Figure 11 we have 3 planes given by the equations x = 3, y = 7
and z = 0. Note that all 3 intersect at the point (3, 7, 0).
Using standard perspective, the set of points whose coordinates satisfy x > 3 are the points in front of the x = 3 plane and
for y > 7 it is the set of points to the right of the plane y = 7.
● Equations of Basic Lines
How do we write the equations of a general line in space? This is a problem we will tackle in a future
section. Here however we exploit again the fact that the coordinate axes meet at right angles to write
the equations of some special lines, namely those that are parallel to coordinate axes. Consider first
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CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
the x-axis. Any point on this axis will have coordinates (x, 0, 0) for some number x ∈ R. Hence we
need two equations to write the equations of the x-axis, namely
y=0 , z=0
(9)
with x unrestricted. What then would be the equations of a line parallel to the x-axis but passing
through the point P = (2, −3, 5)? As before we assume that the x-coordinate must be unrestricted,
but y and z must be restricted to be their current values. Hence equations of this line are
y = −3 , z = 5
(10)
Similar results hold for lines parallel to the y− and z-axes.
●Equations of Spheres
As we did in 2-dimensional space for the circle, we can use the
Euclidean distance formula to set up the standard equation for
a sphere in 3 dimensions. By definition, a sphere of radius R
and center C(h, k, l) is the set of all points a distance R from C.
We let (x, y, z) denote the coordinates of an arbitrary point on
such a sphere as shown in Figure 12. The point with coordinates
(x, y, z) must satisfy Formula (4) using (x1 , y1 , z1 ) = (h, k, l)
and (x2 , y2 , z2 ) = (x, y, z).
√
R = (x − h)2 + (y − k)2 + (z − l)2
The more familiar form of this equation is
Figure 12
(x − h)2 + (y − k)2 + (z − l)2 = R 2
Example 1. Use complete-the-square techniques to determine if the quadratic equation (the largest
exponent on any variable is 2)
x 2 + y 2 + z 2 + 4x − 6z + 3 = 0
is an equation of a sphere. If it is, find the sphere’s center and radius.
Solution: Since y to the first power does not occur in the equation, we need only complete the
square in x and z. Doing this first with x then z we find
x 2 + y2 + z 2 + 4x − 6z + 3 = 0
x 2 + 4x + 4 + y 2 + z 2 − 6z + 3 = 0 + 4
(x + 2)2 + y 2 + z 2 − 6z + 3 = 4
(x + 2)2 + y2 + (z − 3)2 + 3 = 4 + 9
(x + 2)2 + (y − 0)2 + (z − 3)2 = 13 − 3
√ 2
(x + 2)2 + (y − 0)2 + (z − 3)2 = ( 10)
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CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1. CARTESIAN COORDINATES
Hence the given quadratic equation in x, y and z is the equation of a sphere of radius R =
center at (−2, 0, 3).
√
10 with
Example 2. Find an equation of the sphere that has the two points P(2, 5, −3) and Q(4, −9, 14) as
the two endpoints of a diameter of the sphere. Use the fact that the midpoint between any two points
P1 (x1 , y1 , z1 ) and P2 (x2 , y2 , z2 ) is given by averaging the corresponding coordinates:
Midpoint
(
x1 + x2 y1 + y2 z1 + z2
)
,
,
2
2
2
Solution: Figure 13 presents an abstract sketch of the given information.
Figure 13
From this figure we recognize that the coordinates of the center are given by the coordinates of
the midpoint between P and Q. Hence
(h, k, l) = (
2 + 4 5 − 9 −3 + 14
11
) = (3, −2, )
,
,
2
2
2
2
Similarly, the radius R of the sphere will be given by half the distance from P to Q, which is precisely
the distance from either point to the center.
R=
1√
1√
1√
(2 − 4)2 + (5 + 9)2 + (−3 − 14)2 =
4 + 196 + 289 =
489
2
2
2
Hence an equation for the sphere is
2
2
(x − 3) + (y + 2) + (z −
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11 2 489
) =
2
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©2014-16 J. E. Franke, J. R. Griggs and L. K. Norris
CHAPTER 1. EUCLIDEAN GEOMETRY IN 3-DIMENSIONAL SPACE
1.1.2
1.1. CARTESIAN COORDINATES
Exercises
Exercise 1. Sketch on a single set of xyz-axes the points that have coordinates (0, 1, 2), (1, −2, 3) and
(2, 3, −1).
Exercise 2. Find an equation of the plane that is parallel to the xz-plane and is located 14 units to the
left of the xz-plane.
Exercise 3. Find an equation of the plane that passes through the point P0 = (2, 3, 3) and is parallel to
the xy-plane. Sketch the plane.
Exercise 4. Find equations of the line that is parallel to the y-axis and passes through the point
P0 = (2, 3, 3). Sketch the line.
Exercise 5. Describe in words the regions defined by the following inequalities. Sketch the regions.
1. z > 0
2. x y ≥ 0
3. −3 ≤ y ≤ 3
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