Chapter 11
Vectors and the Geometry of Space
Slide - 1
1
Section 11.1
Three-Dimensional
Coordinate Systems
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 2 of 93
Overview




In this chapter we begin the study of multivariable calculus.
To apply calculus in many real-world situations, we introduce
three-dimensional coordinate systems and vectors.
We establish coordinates in space by adding a third axis that
measures distance above and below the xy-plane.
Then we define vectors, which provide simple ways to define
equations for
 lines,
 planes,
 curves, and
 surfaces in space.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 3 of 93
2
Three-Dimensional Coordinate Systems

To locate a point in space, we use three mutually perpendicular
coordinate axes, arranges as in Figure 11.1
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 4 of 93
Three-Dimensional Coordinate Systems

The three coordinate planes 𝑥 = 0, 𝑦 = 0, and 𝑧 = 0 divide space into
eight cells called octants.

The octant in which the point coordinates are all positive is called the first
octant; there is no convention for numbering the other seven octants.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 8 of 93
3
Three-Dimensional Coordinate Systems
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 10 of 93
Three-Dimensional Coordinate Systems
Example 1
We interpret these equations and inequalities geometrically.
a) z ≥ 0
The half-space consisting of the points on and above
the 𝑥𝑦 − 𝑝𝑙𝑎𝑛𝑒.
b) 𝑥 = −3
The plane perpendicular to the x-axis at x = -3.
This plane lies parallel to the yz-plane and 3 units behind it.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 12 of 93
4
Three-Dimensional Coordinate Systems
c)
z = 0, x ≤ 0, y ≥ 0
The second quadrant of the xy-plane.
d) x ≥ 0, y ≥ 0, z ≥ 0
The first quadrant.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 13 of 93
Three-Dimensional Coordinate Systems
e) −1 ≤ 𝑦 ≤ 1
The slab between the planes y = -1
and y = 1 (planes included).
f)
y = −2, z = 2
The line in which the planes y = -2 and z = 2
intersect.
Alternatively, the line through the point (0, -2, 2)
parallel to x-axis.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 14 of 93
5
Three-Dimensional Coordinate Systems
Example 2
What points (x, y, z) satisfy the equations
Solution

Points lie in horizontal plane 𝑧 = 3 and, in this plane, make up the
circle 𝑥 + 𝑦 = 4.

We call this set of points “circle 𝑥 + 𝑦 = 4 in plane 𝑧 = 3” or,
more simply, “circle 𝑥 + 𝑦 = 4, 𝑧 = 3” (Figure 11.4).
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 16 of 93
Distance and Spheres in Space

The formula for the distance between two points in xy-plane
extends to points in space.
Proof
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 17 of 93
6
Distance and Spheres in Space
Example 3
The distance between 𝑃 (2, 1, 5) and𝑃 (−2, 3, 0) is
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 21 of 93
Distance and Spheres in Space

We can use distance formula to write equations for spheres in space
(Figure 11.6).

A point 𝑃(𝑥, 𝑦, 𝑧) lies on sphere of radius 𝑎 centered at 𝑃 (𝑥 , 𝑦 , 𝑧 )
precisely when 𝑃 𝑃 = 𝑎 or
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 23 of 93
7
Distance and Spheres in Space
Example 4
Find the center and radius of the sphere
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 24 of 93
Distance and Spheres in Space
Solution (cont.)
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 25 of 93
8
Distance and Spheres in Space
Example 5
Here are some geometric interpretations of inequalities
and equations involving spheres.
a) 𝑥 + 𝑦 + 𝑧 < 4
The interior of the sphere 𝑥 + 𝑦 + 𝑧 = 4.
b) 𝑥 + 𝑦 + 𝑧 ≤ 4
Solid ball bounded by sphere 𝑥 + 𝑦 + 𝑧 = 4.
Alternatively, the sphere 𝑥 + 𝑦 + 𝑧 = 4
together with its interior.
c) 𝑥 + 𝑦 + 𝑧 > 4
The exterior of sphere 𝑥 + 𝑦 + 𝑧 = 4.
d) 𝑥 + 𝑦 + 𝑧 = 4, 𝑧 ≤ 0
Lower hemisphere cut from the sphere 𝑥 + 𝑦 + 𝑧 = 4
by xy-plane (the plane 𝑧 = 0).
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 28 of 93
End of
Section 11.1 Three-Dimensional
Coordinate Systems
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 30 of 93
9
Section 11.2
Vectors
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 31 of 93
Introduction







Some of things we measure are determined simply by their
magnitudes.
To record mass, length, or time, for example, we need only write down
number and name appropriate unit of measure.
We need more information to describe a force, displacement, or
velocity.
To describe force, we need to record direction in which it acts as well
as how large it is.
To describe body’s displacement, we have to say in what direction it
moved as well as how far.
To describe body’s velocity, we have to know where body is headed as
well as how fast it is going.
In this section we show how to represent things that have both
magnitude and direction in plane or in space.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 32 of 93
10
Component Form

Quantity such as force, displacement, or velocity is called vector and is
represented by directed line segment.

Arrow points in direction of action and its
length gives magnitude of action in terms of suitably chosen unit.

For example, force vector points in direction in which force acts and its
length is measure of force’s strength;
velocity vector points in direction of motion and
its length is speed of moving object.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 33 of 93
Component Form

Figure 11.8 displays velocity vector 𝒗 at specific location for
particle moving along a path in plane or in space. (This
application of vectors is studied in Chapter 12.)
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 34 of 93
11
Component Form
DEFINITIONS

Vector represented by directed line segment 𝐴𝐵 has




initial point A and
terminal point B and
its length is denoted by 𝐴𝐵
.
Two vectors are equal if they have the same length and direction.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 35 of 93
Component Form
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 39 of 93
12
Component Form
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 40 of 93
Component Form
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 41 of 93
13
Component Form
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 43 of 93
Component Form
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 44 of 93
14
Component Form
Example 2 A small cart is being pulled along a smooth horizontal floor
with a 20-N force 𝐹 making a 45° angle to floor Figure 11.11 .
What is the effective force moving the cart forward?
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 46 of 93
Vector Algebra Operations



Two principal operations involving vectors are
 vector addition and
 scalar multiplication.
Scalar is simply real number, and is called such when we want to
draw attention to the differences between number and vectors.
Scalars can be positive, negative, or zero and are used to “scale” a
vector by multiplication.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 48 of 93
15
Vector Algebra Operations


Definition of vector addition is illustrated
geometrically for planar vectors, where initial
point of one vector is placed at terminal point
of the other.
Another interpretation is called
parallelogram law of addition, where sum,
called resultant vector, is diagonal of
parallelogram.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 50 of 93
Vector Algebra Operations
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 52 of 93
16
Vector Algebra Operations
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 53 of 93
Vector Algebra Operations
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 54 of 93
17
Vector Algebra Operations

Vector operations have many of properties of ordinary arithmetic.

These properties are readily verified using definitions of vector
addition and multiplication by a scalar.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 56 of 93
Vector Algebra Operations

For instance, to establish Property 1, we have
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 57 of 93
18
Unit Vectors
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 59 of 93
Unit Vectors

As shown in Figure 11.15, the component form for the vector
from 𝑃 (𝑥 , 𝑦 , 𝑧 ) to 𝑃 (𝑥 , 𝑦 , 𝑧 ) is
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 60 of 93
19
Unit Vectors

If v ≠ 0, then its length v is not zero and
1
1
v = 1.
v =
v
v

That is
𝐯
𝐯
is unit vector in the direction of v, called the direction
of the nonzero vector v.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 61 of 93
Unit Vectors
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 62 of 93
20
Unit Vectors
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 64 of 93
Unit Vectors

In summary, we can express any nonzero vector v in terms of its
two important features, length and direction, by writing
v= v
.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 66 of 93
21
Unit Vectors
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 67 of 93
Midpoint of Line Segment


Vectors are often useful in geometry.
For example, coordinates of midpoint of a line segment are
found by averaging.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 69 of 93
22
Midpoint of Line Segment

To see why, observe (Figure 11.16) that
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 70 of 93
Midpoint of a Line Segment
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 72 of 93
23
Applications
Example 8
A jet airliner, flying due east at 800 km/h in still air, encounters a 110 km/h
tailwind blowing in the direction 60° north of east.
The airplane holds its compass heading due east but, because of the wind,
acquires a new ground speed and direction. What are they?
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 74 of 93
Applications
Solution
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 75 of 93
24
Applications
Solution (cont.)
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 76 of 93
Applications
Example 9
A 75-N weight is suspended by two wires, as shown in Figure 11.18a.
Find the forces F and F acting in both wires.
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 80 of 93
25
Solution
Applications
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 81 of 93
Solution (cont.)
Applications
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 82 of 93
26
End of
Section 11.2 Vectors
Thomas' Calculus, 14e in SI Units
Copyright © 2020 Pearson Education Ltd.
Slide 87 of 93