7-4 Algebraic Vectors
SECTION
7-4
531
Algebraic Vectors
•
•
•
•
•
From Geometric Vectors to Algebraic Vectors
Vector Addition and Scalar Multiplication
Unit Vectors
Algebraic Properties
Static Equilibrium
Geometric vectors in a plane are readily generalized to three-dimensional space. However, to generalize vectors further to higher-dimensional abstract spaces, it is essential to define the vector concept algebraically. This is done in such a way that the geometric vectors become special cases of the more general algebraic vectors. Algebraic
vectors have many advantages over geometric vectors. One advantage will become
apparent when we consider static equilibrium problems at the end of the section.
The development of algebraic vectors in this book is introductory in nature and
is restricted to the plane. Further study of vectors in three- and higher-dimensional
spaces is reserved for more advanced mathematical courses.
• From Geometric
Vectors to
Algebraic Vectors
y
B
The transition from geometric vectors to algebraic vectors is begun by placing geometric vectors in a rectangular coordinate system. A geometric vector AB in a rectangular coordinate system translated so that its initial point is at the origin is said to
such that OP
be in standard position. The vector OP
AB is said to be the stan
dard vector for AB (see Fig. 1).
in Figure 1 is the standard vector for infinitely many
Note that the vector OP
.
vectors—all vectors with the same magnitude and direction as OP
P
Standard
vector
A
O
x
is the standard
FIGURE 1 OP
vector for AB .
EXPLORE-DISCUSS 1
as their stan(A) In a copy of Figure 1, draw in three other vectors having OP
dard vector.
(B) If the tail of a vector is at point A(3, 2) and its tip is at B(6, 4), discuss how
is the standard vector for you would find the coordinates of P so that OP
AB .
Given the coordinates of the endpoints of a geometric vector in a rectangular
coordinate system, how do we find its corresponding standard vector? The process is
are always (0, 0). Thus,
not difficult. The coordinates of the initial point, O, of OP
. The coordinates
we have only to find the coordinates of P, the terminal point of OP
of P are given by
(xp, yp) (xb xa, yb ya)
(1)
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7 Additional Topics in Trigonometry
where the coordinates of A are (xa, ya) and the coordinates of B are (xb, yb). Example
1 illustrates the use of equation (1).
EXAMPLE 1
Finding a Standard Vector for a Given Vector
Given the geometric vector AB with initial point A(3, 4) and terminal point B(7, 1),
for find the standard vector OP
AB . That is, find the coordinates of the point P such
that OP
AB .
Solution
y
The coordinates of P are given by
(xp, yp) (xb xa, yb ya)
A(3, 4)
(7 3, 1 4)
O
x
B(7, 1)
Standard
vector
P(4, 5)
(4, 5)
Note in Figure 2 that if we start at A, then move to the right 4 units and down 5 units,
we will be at B. If we start at the origin, then move to the right 4 units and down
5 units, we will be at P.
FIGURE 2
Matched Problem 1
y
P(a, b)
v a, b
O
x
FIGURE 3 Algebraic vector a, b
associated with a geometric vector
.
OP
Given the geometric vector AB with initial point A(8, 3) and terminal point B(4, 5),
for find the standard vector OP
AB .
The preceding discussion suggests another way of looking at vectors. Since, given
any geometric vector AB in a rectangular coordinate system, there always exists a
point P(xp, yp) such that OP
AB , the point P(xp, yp) completely specifies the vector AB , except for its position. And we are not concerned about its position because
we are free to translate AB anywhere we please. Conversely, given any point P(xp, yp)
in a rectangular coordinate system, the directed line segment joining O to P forms
.
the geometric vector OP
This leads us to define an algebraic vector as an ordered pair of real numbers.
To avoid confusing a point (a, b) with a vector (a, b), we use a, b to represent an
algebraic vector. Geometrically, the algebraic vector a, b corresponds to the stan with terminal point P(a, b) and initial point O(0, 0), as
dard (geometric) vector OP
illustrated in Figure 3.
The real numbers a and b are scalar components of the vector a, b. The word
scalar means real number and is often used in the context of vectors where one refers
to “scalar quantities” as opposed to “vector quantities.” Thus, we talk about “scalar
components” and “vector components” of a given vector. The words “scalar” and
“vector” are often dropped if the meaning of component is clear from the context.
Two vectors u a, b and v c, d are said to be equal if their corresponding components are equal, that is, if a c and b d. The zero vector is denoted
by 0 0, 0.
Geometric vectors are limited to spaces we can visualize, that is, to two- and
three-dimensional spaces. Algebraic vectors do not have these restrictions. The following are algebraic vectors from two-, three-, four-, and five-dimensional spaces:
7-4 Algebraic Vectors
2, 5
3, 0, 8
5, 1, 1, 2
533
1, 0, 1, 3, 4
As we said earlier, the discussion in this book is limited to algebraic vectors in a twodimensional space, which represents a plane.
We now define the magnitude of an algebraic vector:
DEFINITION 1
Magnitude of v a, b
The magnitude, or norm, of a vector v a, b is denoted by v and is given
by
y
P(a, b)
v a2 b2
v a 2 b 2
O
x
FIGURE 4 Magnitude of vector
a, b geometrically interpreted.
EXAMPLE 2
assoGeometrically, a2 b2 is the length of the standard geometric vector OP
ciated with the algebraic vector a, b (see Fig. 4).
The definition of magnitude is readily generalized to higher-dimensional vector
spaces. For example, if v a, b, c, d, then the magnitude, or norm, is given by
a2 b2 c2 d2. But now we are not able to interpret the result in terms of geometric vectors.
Finding the Magnitude of a Vector
Find the magnitude of the vector v 3, 5.
v 32 (5)2 34
Solution
Matched Problem 2
• Vector Addition
and Scalar
Multiplication
DEFINITION 2
Find the magnitude of the vector v 2, 4.
To add two algebraic vectors, add the corresponding components as indicated in the
following definition of addition:
Vector Addition
If u a, b and v c, d, then
u v a c, b d
The definition of addition of algebraic vectors is consistent with the parallelogram and tail-to-tip definitions for adding geometric vectors given in Section 7-3 (see
Explore-Discuss 2).
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7 Additional Topics in Trigonometry
EXPLORE-DISCUSS 2
If u 3, 2, v 7, 3, then u v 3 7, 2 3 4, 5. Locate u,
v, and u v in a rectangular coordinate system and interpret geometrically in
terms of the parallelogram and tail-to-tip rules discussed in the last section.
To multiply a vector by a scalar (a real number) multiply each component by the
scalar:
DEFINITION 3
Scalar Multiplication
If u a, b and k is a scalar, then
ku ka, b ka, kb
Geometrically, if a vector v is multiplied by a scalar k, the magnitude of the vector v is multiplied by k. If k is positive, then kv has the same direction as v. If k is
negative, then kv has the opposite direction as v. These relationships are illustrated
in Figure 5.
2v
v
v
0.5v
FIGURE 5 Scalar multiplication
geometrically interpreted.
EXAMPLE 3
Vector Addition and Scalar Multiplication
Let u 4, 3, v 2, 3, and w 0, 5, find:
(A) u v
Solutions
(B) 2u
(C) 2u 3v
(D) 3u 2v w
(A) u v 4, 3 2, 3 6, 0
(B) 2u 24, 3 8, 6
(C) 2u 3v 24, 3 32, 3
8, 6 6, 9 2, 15
(D) 3u 2v w 34, 3 22, 3 0, 5
12, 9 4, 6 0, 5
16, 2
Matched Problem 3
Let u 5, 3, v 4, 6, and w 2, 0, find:
(A) u v
(B) 3u
(C) 3u 2v
(D) 2u v 3w
7-4 Algebraic Vectors
• Unit Vectors
If v 1, then v is called a unit vector. A unit vector can be formed from an arbitrary nonzero vector as follows:
A Unit Vector with the Same Direction as v
If v is a nonzero vector, then
u
1
v
v
is a unit vector with the same direction as v.
EXAMPLE 4
Finding a Unit Vector with the Same Direction as a Given Vector
Given a vector v 1, 2, find a unit vector u with the same direction as v.
v 12 (2)2 5
Solution
u
1
1
v
1, 2
v
5
2
51 , 5
u Check
1 4
5 5 1 1
1
5
2
2
5
2
And we see that u is a unit vector with the same direction as v.
Matched Problem 4
535
Given a vector v 3, 1, find a unit vector u with the same direction as v.
We now define two very important unit vectors, the i and j unit vectors.
The i and j Unit Vectors
y
1
i 1, 0
j
j 0, 1
0
i
x
1
536
7 Additional Topics in Trigonometry
Why are the i and j unit vectors so important? One of the reasons is that any
vector v a, b can be expressed as a linear combination of those two vectors; that
is, as ai bj.
v a, b a, 0 0, b
a1, 0 b0, 1 ai bj
EXAMPLE 5
Expressing a Vector in Terms of the i and j Vectors
Express each vector as a linear combination of the i and j unit vectors.
(A) 2, 4
Solutions
Matched Problem 5
Properties
(C) 0, 7
(A) 2, 4 2i 4j
(B) 2, 0 2i 0j 2i
(C) 0, 7 0i 7j 7j
Express each vector as a linear combination of the i and j unit vectors.
(A) 5, 3
• Algebraic
(B) 2, 0
(B) 9, 0
(C) 0, 6
Vector addition and scalar multiplication possess algebraic properties similar to the
real numbers. These properties enable us to manipulate symbols representing vectors
and scalars in much the same way we manipulate symbols that represent real numbers in algebra. These properties are listed below for convenient reference.
Algebraic Properties of Vectors
A. Addition Properties. For all vectors u, v, and w:
1.
2.
3.
4.
u
u
u
u
vv
u
(v w) (u v) w
00
uu
(u) (u) u 0
Commutative Property
Associative Property
Additive Identity
Additive Inverse
B. Scalar Multiplication Properties. For all vectors u and v and all scalars
m and n:
1.
2.
3.
4.
m(nu) (mn)u
m(u v) mu mv
(m n)u mu nu
1u u
Associative Property
Distributive Property
Distributive Property
Multiplicative Identity
7-4 Algebraic Vectors
EXAMPLE 6
537
Algebraic Operations on Vectors Expressed in Terms of the i and j Vectors
For u i 2j and v 5i 2j, compute each of the following:
(A) u v
Solutions
(B) u v
(C) 2u 3v
(A) u v (i 2j) (5i 2j)
i 2j 5i 2j 6i 0j 6i
(B) u v (i 2j) (5i 2j)
i 2j 5i 2j 4i 4j
(C) 2u 3v 2(i 2j) 3(5i 2j)
2i 4j 15i 6j 17i 2j
Matched Problem 6
For u 2i j and v 4i 5j, compute each of the following:
(A) u v
• Static Equilibrium
(B) u v
(C) 3u 2v
Algebraic vectors can be used to solve many types of problems in physics and engineering. We complete this section by considering a few problems involving static
equilibrium. Fundamental to our approach are two basic principles regarding forces
and objects subject to these forces:
Conditions for Static Equilibrium
1. An object at rest is said to be in static equilibrium.
2. For an object located at the origin in a rectangular coordinate system to
remain in static equilibrium, at rest, it is necessary that the sum of all the
force vectors acting on the object be the zero vector.
Example 7 shows how some important physics/engineering problems can be
solved using algebraic vectors and the conditions for static equilibrium. It is assumed
that you know how to solve a system of two equations with two variables. In case
you need a reminder, procedures are reviewed in Section 1-2.
EXAMPLE 7
Tension in Cables
A cable car, used to ferry people and supplies across a river, weighs 2,500 pounds
fully loaded. The car stops when partway across and deflects the cable relative to the
538
7 Additional Topics in Trigonometry
horizontal, as indicated in Figure 6. What is the tension in each part of the cable running to each tower?
FIGURE 6
15
7
2,500 pounds
River
Solution
Step 1. Draw a force diagram with all force vectors in standard position at the
origin (Fig. 7). The objective is to find u and v.
Step 2. Write each force vector in terms of the i and j unit vectors:
y
u u(cos 7°)i u(sin 7°)j
v
7
v v(cos 15°)i v(sin 15°)j
u
x
15
w 2,500j
w
w 2,500 pounds
FIGURE 7
Step 3. For the system to be in static equilibrium, the sum of the force vectors
must be the zero vector. That is,
u
v
w0
Replacing vectors u, v, and w from step 2, we obtain
[u(cos 7°)i u(sin 7°)j] [v(cos 15°)i v(sin 15°)j] 2,500j 0i 0j
which on combining i and j vectors becomes
[u(cos 7°) v(cos 15°)]i [u(sin 7°) v(sin 15°) 2,500]j 0i 0j
Since two vectors are equal if and only if their corresponding components are equal, we are led to the following system of two equations
in the two variables u and v:
(cos 7°)u (cos 15°)v
(sin 7°)u 0
(sin 15°)v 2,500 0
Solving this system by standard methods, we find that
u 6,400 pounds
and
v 6,600 pounds
Did you expect that the tension in each part of the cable is more than
the weight hanging from the cable?
7-4 Algebraic Vectors
Matched Problem 7
539
Repeat Example 7 with 15° replaced with 13°, 7° replaced with 9°, and the 2,500
pounds replaced with 1,900 pounds.
Answers to Matched Problems
1.
3.
4.
5.
6.
7.
EXERCISE
P(4, 8)
2. 25
(A) 1, 3
(B) 15, 9
(C) 23, 21
u 3/10, 1/10
(A) 5i 3j
(B) 9i
(C) 6j
(A) 6i 4j
(B) 2i 6j
(C) 2i 13j
u 4,900 lb, v 5,000 lb
(D) 20, 12
7-4
A
23. u v
24. u v
25. 2u 3v
In Problems 1–6, represent each geometric vector AB , with
endpoints as indicated, as an algebraic vector in the form
a, b.
26. 3u 2v
27. 2u v 2w
28. u 3v 2w
1. A(0, 0), B(7, 2)
2. A(5, 3), B(0, 0)
3. A(4, 0), B(0, 8)
4. A(0, 5), B(6, 0)
5. A(9, 4), B(7, 5)
6. A(6, 3), B(9, 1)
In Problems 7–12, find the magnitude of each vector.
7. 15, 0
10. 48, 20
8. 0, 32
9. 21, 72
11. 155, 468
12. 836, 123
29. v 1, 1
30. v 2, 1
31. v 12, 5
32. v 7, 24
In Problems 33–36, determine whether the statement is true
or false. If true, explain why. If false, give a counterexample.
33. If u is a scalar multiple of v, then u and v have the same direction.
34. If u and v are nonzero vectors that have the same direction,
then u is a scalar multiple of v.
B
35. The sum of two unit vectors is a unit vector.
In Problems 13–16, find:
(A) u v
In Problems 29–32, find a unit vector u with the same direction as v.
(B) u v
(C) 2u v 3w
36. If u is a unit vector and k is a scalar, then the magnitude of
ku is k.
13. u 2, 1, v 1, 3, w 3, 0
14. u 1, 2, v 3, 2, w 0, 2
15. u 4, 1, v 2, 2, w 0, 1
16. u 3, 2, v 2, 2, w 3, 0
In Problems 17–22, express v in terms of the i and j unit
vectors.
17. 8, 0
18. 0, 14
19. 6, 12
20. 5, 18
21. v AB , where A (2, 3) and B (3, 1)
22. v AB , where A (2, 1) and B (0, 2)
In Problems 23–28, let u 3i 2j, v 2i 4j, and
w 2i, and perform the indicated operations.
C
In Problems 37–44, let u a, b, v c, d, and w e, f be vectors and m and n be scalars. Prove each of the following vector properties using appropriate properties of real
numbers and the definitions of vector addition and scalar
multiplication.
37. u (v w) (u v) w
38. u v v u
39. u 0 u
40. u (u) 0
41. (m n)u m u nu
42. m(u v) m u mv
43. m (nu) (mn)u
44. 1u u
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7 Additional Topics in Trigonometry
APPLICATIONS
In Problems 45–52, compute all answers to 3 significant
digits.
45. Static Equilibrium. A unicyclist at a certain point on a
tightrope deflects the rope as indicated in the figure. If the
total weight of the cyclist and the unicycle is 155 pounds,
how much tension is in each part of the cable?
49. Static Equilibrium. A 400-pound sign is suspended as
shown in figure (a). The corresponding force diagram (b) is
formed by observing the following: Member AB is “pushing” at B and is under compression. This “pushing” force
also can be thought of as the force vector a “pulling” to the
right at B. The force vector b reflects the fact that member
CB is under tension—that is, it is “pulling” at B. The force
vector c corresponds to the weight of the sign “pulling”
down at B. Find the magnitudes of the forces in the rigid
supporting members; that is, find a and b in the force
diagram (b).
C
y
2 yards
6.2
5.5
155 pounds
b
A
a
1 yard
B
x
46. Static Equilibrium. Repeat Problem 45 with the left angle
4.2°, the right angle 5.3°, and the total weight 112 pounds.
47. Static Equilibrium. A weight of 1,000 pounds is suspended from two cables as shown in the figure. What is the
tension in each cable?
45.0
30.0
c
400 pounds
(a)
(b)
50. Static Equilibrium. A weight of 1,000 kilograms is supported as shown in the figure. What are the magnitudes of
the forces on the members AB and BC?
C
1,000
pounds
2 meters
1 meter
48. Static Equilibrium. A weight of 500 pounds is supported by
two cables as illustrated. What is the tension in each cable?
B
A
1000
kilograms
45.0
20.0
500
pounds
51. Static Equilibrium. A 1,250-pound weight is hanging
from a hoist as indicated in the figure on the next page.
What are the magnitudes of the forces on the members AB
and BC?
7-5
C
10.6 feet
Polar Coordinates and Graphs
541
52. Static Equilibrium. A weight of 5,000 kilograms is supported as shown in the figure. What are the magnitudes of
the forces on the members AB and BC?
B
12.5
feet
C
A
5 meters
B
6m
5,000
kilograms
A
1,250
pounds
Figure for 51
Figure for 52
SECTION
7-5
Polar Coordinates and Graphs
•
•
•
•
•
Polar Coordinate System
Converting from Polar to Rectangular Form, and Vice Versa
Graphing Polar Equations
Some Standard Polar Curves
Application
Up until now we have used only the rectangular coordinate system. Other coordinate
systems have particular advantages in certain situations. Of the many that are possible, the polar coordinate system ranks second in importance to the rectangular coordinate system and forms the subject matter for this section.
• Polar Coordinate
System
Pole
Polar axis
O
P(r, )
r
O
FIGURE 1 Polar coordinate system.
To form a polar coordinate system in a plane (see Fig. 1), start with a fixed point
O and call it the pole, or origin. From this point draw a half line, or ray (usually horizontal and to the right), and call this line the polar axis.
If P is an arbitrary point in a plane, then associate polar coordinates (r, ) with
it as follows: Starting with the polar axis as the initial side of an angle, rotate the terminal side until it, or the extension of it through the pole, passes through the point.
The coordinate in (r, ) is this angle, in degree or radian measure. The angle is
positive if the rotation is counterclockwise and negative if the rotation is clockwise.
The r coordinate in (r, ) is the directed distance from the pole to the point P. It is
positive if measured from the pole along the terminal side of and negative if measured along the terminal side extended through the pole.
Figure 2 illustrates a point P with three different sets of polar coordinates. Study
this figure carefully. The pole has polar coordinates (0, ) for arbitrary . For example, (0, 0°), (0, /3), and (0, 371°) are all coordinates of the pole.
FIGURE 2 Polar coordinates of a
point.
4
5
5
5
P
P
4, 4 225
P
(4, 225)
5
5
5
(a)
3
4
(b)
(c)
4, 34