Chapter 2 Supplementary Exercises
3.1 Vectors in 2-Space, 3-Space, and n-Space
Linear algebra is primarily concerned with two types of mathematical objects, “matrices”
and “vectors.” In Chapter 1 we discussed the basic properties of matrices, we introduced the idea of viewingn-tuples of real numbers as vectors, and we denoted the set of all such n-tuples asRn. In this section we will review the basic properties of vectors in two and three dimensions with the goal of extending these properties to vectors inRn.
Geometric Vectors Engineers and physicists represent vectors in two dimensions (also called2-space) or in three dimensions (also called 3-space) by arrows. The direction of the arrowhead specifies thedirectionof the vector and thelengthof the arrow specifies the magnitude.
Mathematicians call thesegeometric vectors. The tail of the arrow is called theinitial pointof the vector and the tip theterminal point(Figure 3.1.1).
Terminal point
Initial point Figure 3.1.1
In this text we will denote vectors in boldface type such asa,b,v,w, andx, and we will denote scalars in lowercase italic type such asa,k,v,w, andx. When we want to indicate that a vectorvhas initial pointAand terminal pointB, then, as shown in Figure 3.1.2, we will write
v=−→
AB
Vectors with the same length and direction, such as those in Figure 3.1.3, are said to beequivalent. Since we want a vector to be determined solely by its length and direction, equivalent vectors are regarded as the same vector even though they may be in different positions. Equivalent vectors are also said to beequal, which we indicate by writing
v=w
v = AB v
B
A
Figure 3.1.2
Equivalent vectors Figure 3.1.3
The vector whose initial and terminal points coincide has length zero, so we call this thezero vectorand denote it by0. The zero vector has no natural direction, so we will agree that it can be assigned any direction that is convenient for the problem at hand.
Vector Addition There are a number of important algebraic operations on vectors, all of which have their origin in laws of physics.
Parallelogram Rule for Vector Addition Ifvandware vectors in 2-space or 3-space that are positioned so their initial points coincide, then the two vectors form adjacent sides of a parallelogram, and thesum v+wis the vector represented by the arrow from the common initial point ofvandwto the opposite vertex of the parallelogram (Figure 3.1.4a).
Here is another way to form the sum of two vectors.
Triangle Rule for Vector Addition Ifvandware vectors in 2-space or 3-space that are positioned so the initial point ofwis at the terminal point ofv, then thesumv+w is represented by the arrow from the initial point of vto the terminal point of w (Figure 3.1.4b).
In Figure 3.1.4cwe have constructed the sumsv+wandw+vby the triangle rule.
This construction makes it evident that
v+w=w+v (1)
and that the sum obtained by the triangle rule is the same as the sum obtained by the parallelogram rule.
Figure 3.1.4
v v + w
w
(b)
v v
w + v v + w
w
w (c) v v + w
w (a)
Vector addition can also be viewed as a process of translating points.
Vector Addition Viewed asTranslation Ifv,w, andv+ware positioned so their initial points coincide, then the terminal point ofv+wcan be viewed in two ways:
1. The terminal point ofv+w is the point that results when the terminal point ofvis translated in the direction of w by a distance equal to the length ofw (Figure 3.1.5a).
2. The terminal point ofv+w is the point that results when the terminal point ofw is translated in the direction of v by a distance equal to the length ofv (Figure 3.1.5b).
Accordingly, we say that v+w is the translation of v by w or, alternatively, the translation ofwbyv.
3.1 Vectors in 2-Space, 3-Space, andn-Space 133
Figure 3.1.5
v v + w
w (b) v v + w
w (a)
Vector Subtraction In ordinary arithmetic we can writea−b=a+(−b), which expresses subtraction in terms of addition. There is an analogous idea in vector arithmetic.
Vector Subtraction Thenegativeof a vectorv, denoted by−v, is the vector that has the same length asvbut is oppositely directed (Figure 3.1.6a), and thedifferenceofv fromw, denoted byw−v, is taken to be the sum
w−v=w+(−v) (2)
The difference of v from w can be obtained geometrically by the parallelogram method shown in Figure 3.1.6b, or more directly by positioningwandvso their ini- tial points coincide and drawing the vector from the terminal point ofvto the terminal point ofw(Figure 3.1.6c).
Figure 3.1.6
–v v
–v v
w – v w
(a) (b)
v w
w – v
(c)
Scalar Multiplication Sometimes there is a need to change the length of a vector or change its length and reverse its direction. This is accomplished by a type of multiplication in which vectors are multiplied by scalars. As an example, the product 2vdenotes the vector that has the same direction asvbut twice the length, and the product−2vdenotes the vector that is oppositely directed tovand has twice the length. Here is the general result.
Scalar Multiplication If vis a nonzero vector in 2-space or 3-space, and if k is a nonzero scalar, then we define thescalar product ofvbykto be the vector whose length is|k|times the length ofvand whose direction is the same as that ofvifkis positive and opposite to that ofvifkis negative. Ifk=0 orv=0, then we definekv to be0.
Figure 3.1.7 shows the geometric relationship between a vectorvand some of its scalar multiples. In particular, observe that(−1)vhas the same length asvbut is oppositely directed; therefore,
(−1)v= −v (3)
v
2v
(–1)v
(–3)v
1v
2
Figure 3.1.7
Parallel and Collinear Vectors
Suppose thatvandware vectors in 2-space or 3-space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a common line, so it is reasonable to say that they arecollinear(Figure 3.1.8a). However, if we trans- late one of the vectors, as indicated in Figure 3.1.8b, then the vectors areparallel but no longer collinear. This creates a linguistic problem because translating a vector does not change it. The only way to resolve this problem is to agree that the termsparalleland
collinearmean the same thing when applied to vectors. Although the vector0has no clearly defined direction, we will regard it as parallel to all vectors when convenient.
Figure 3.1.8
v
kv v
kv
(a) (b)
Sums ofThree or More Vectors
Vector addition satisfies theassociative law for addition, meaning that when we add three vectors, sayu,v, andw, it does not matter which two we add first; that is,
u+(v+w)=(u+v)+w
It follows from this that there is no ambiguity in the expressionu+v+wbecause the same result is obtained no matter how the vectors are grouped.
A simple way to constructu+v+wis to place the vectors “tip to tail” in succession and then draw the vector from the initial point ofuto the terminal point ofw(Figure 3.1.9a). The tip-to-tail method also works for four or more vectors (Figure 3.1.9b).
The tip-to-tail method makes it evident that ifu,v, andware vectors in 3-space with a common initial point, thenu+v+wis the diagonal of the parallelepiped that has the three vectors as adjacent sides (Figure 3.1.9c).
Figure 3.1.9
u v
x
u + v + w + x w u
v
w u + v
v + u + (v + w) w
(u + v) + w u
u + v + w v w
(a) (b) (c)
Vectors in Coordinate Systems
Up until now we have discussed vectors without reference to a coordinate system. How- ever, as we will soon see, computations with vectors are much simpler to perform if a coordinate system is present to work with.
If a vectorvin 2-space or 3-space is positioned with its initial point at the origin of a rectangular coordinate system, then the vector is completely determined by the coor- dinates of its terminal point (Figure 3.1.10). We call these coordinates thecomponents The component forms of the
zero vector are 0=(0,0) in 2-space and0=(0,0,0)in 3- space.
ofvrelative to the coordinate system. We will writev=(v1, v2)to denote a vectorvin 2-space with components(v1, v2), andv=(v1, v2, v3)to denote a vectorvin 3-space with components(v1, v2, v3).
Figure 3.1.10
y z
x
v
(v1, v2, v3) v
(v1, v2)
x y
3.1 Vectors in 2-Space, 3-Space, andn-Space 135 It should be evident geometrically that two vectors in 2-space or 3-space are equiv- alent if and only if they have the same terminal point when their initial points are at the origin. Algebraically, this means that two vectors are equivalent if and only if their corresponding components are equal. Thus, for example, the vectors
v=(v1, v2, v3) and w=(w1, w2, w3) in 3-space are equivalent if and only if
v1=w1, v2=w2, v3=w3
Remark It may have occurred to you that an ordered pair(v1, v2)can represent either a vector withcomponentsv1andv2or a point withcoordinatesv1andv2(and similarly for ordered triples).
x y
(v1, v2)
Figure 3.1.11 The ordered pair(v1, v2)can represent a point or a vector.
Both are valid geometric interpretations, so the appropriate choice will depend on the geometric viewpoint that we want to emphasize (Figure 3.1.11).
Vectors Whose Initial Point Is Not at the Origin
It is sometimes necessary to consider vectors whose initial points are not at the origin.
If−−→
P1P2denotes the vector with initial pointP1(x1, y1)and terminal pointP2(x2, y2), then the components of this vector are given by the formula
−−→P1P2=(x2−x1, y2−y1) (4)
That is, the components of−−→
P1P2 are obtained by subtracting the coordinates of the initial point from the coordinates of the terminal point. For example, in Figure 3.1.12
v = P1P2 = OP2 – OP1 x y
v OP2 OP1
P1(x1, y1)
P2(x2, y2)
Figure 3.1.12
the vector−−→
P1P2is the difference of vectors−−→
OP2and−−→
OP1, so
−−→P1P2=−−→
OP2−−−→
OP1=(x2, y2)−(x1, y1)=(x2−x1, y2−y1)
As you might expect, the components of a vector in 3-space that has initial point P1(x1, y1, z1)and terminal pointP2(x2, y2, z2)are given by
−−→P1P2=(x2−x1, y2−y1, z2−z1) (5)
E X A M P L E 1 Finding the Components of a Vector The components of the vectorv=−−→
P1P2 with initial pointP1(2,−1,4)and terminal pointP2(7,5,−8)are
v=(7−2,5−(−1), (−8)−4)=(5,6,−12)
n-Space The idea of using ordered pairs and triples of real numbers to represent points in two- dimensional space and three-dimensional space was well known in the eighteenth and nineteenth centuries. By the dawn of the twentieth century, mathematicians and physi- cists were exploring the use of “higher dimensional” spaces in mathematics and physics.
Today, even the layman is familiar with the notion of time as a fourth dimension, an idea used by Albert Einstein in developing the general theory of relativity. Today, physicists working in the field of “string theory” commonly use 11-dimensional space in their quest for a unified theory that will explain how the fundamental forces of nature work. Much of the remaining work in this section is concerned with extending the notion of space to ndimensions.
To explore these ideas further, we start with some terminology and notation. The set of all real numbers can be viewed geometrically as a line. It is called thereal lineand is denoted byR orR1. The superscript reinforces the intuitive idea that a line is one- dimensional. The set of all ordered pairs of real numbers (called2-tuples) and the set of all ordered triples of real numbers (called3-tuples) are denoted byR2andR3, respectively.
The superscript reinforces the idea that the ordered pairs correspond to points in the plane (two-dimensional) and ordered triples to points in space (three-dimensional). The following definition extends this idea.
DEFINITION1 Ifnis a positive integer, then anordered n-tupleis a sequence ofn real numbers(v1, v2, . . . , vn). The set of all orderedn-tuples is calledn-spaceand is denoted byRn.
Remark You can think of the numbers in ann-tuple(v1, v2, . . . , vn)as either the coordinates of ageneralized pointor the components of ageneralized vector, depending on the geometric image you want to bring to mind—the choice makes no difference mathematically, since it is the algebraic properties ofn-tuples that are of concern.
Here are some typical applications that lead ton-tuples.
• Experimental Data—A scientist performs an experiment and makes n numerical measurements each time the experiment is performed. The result of each experiment can be regarded as a vectory=(y1, y2, . . . , yn)inRn in whichy1, y2, . . . , yn are the measured values.
• Storage and Warehousing—A national trucking company has 15 depots for storing and servicing its trucks. At each point in time the distribution of trucks in the service depots can be described by a 15-tuplex=(x1, x2, . . . , x15)in whichx1is the number of trucks in the first depot,x2is the number in the second depot, and so forth.
• Electrical Circuits—A certain kind of processing chip is designed to receive four input voltages and produce three output voltages in response. The input voltages can be regarded as vectors inR4and the output voltages as vectors inR3. Thus, the chip can be viewed as a device that transforms an input vectorv=(v1, v2, v3, v4)in R4into an output vectorw=(w1, w2, w3)inR3.
• Graphical Images—One way in which color images are created on computer screens is by assigning each pixel (an addressable point on the screen) three numbers that describe thehue,saturation, andbrightnessof the pixel. Thus, a complete color image can be viewed as a set of 5-tuples of the formv=(x, y, h, s, b)in whichxandyare the screen coordinates of a pixel andh, s, andbare its hue, saturation, and brightness.
• Economics—One approach to economic analysis is to divide an economy into sectors (manufacturing, services, utilities, and so forth) and measure the output of each sector by a dollar value. Thus, in an economy with 10 sectors the economic output of the entire economy can be represented by a 10-tuples=(s1, s2, . . . , s10)in which the numberss1, s2, . . . , s10are the outputs of the individual sectors.
Albert Einstein (1879–1955)
Historical Note The German-born physicist Albert Einstein immigrated to the United States in 1935, where he settled at Princeton University. Einstein spent the last three decades of his life working unsuccessfully at producing aunified field theorythat would establish an underlying link between the forces of gravity and electromagnetism. Recently, physi- cists have made progress on the problem using a frame- work known asstring theory. In this theory the smallest, indivisible components of the Universe are not particles but loops that behave like vibrating strings. Whereas Einstein’s space-time universe was four-dimensional, strings reside in an 11-dimensional world that is the focus of current re- search.
[Image: © Bettmann/CORBIS]
3.1 Vectors in 2-Space, 3-Space, andn-Space 137
• Mechanical Systems—Suppose that six particles move along the same coordinate line so that at time t their coordinates are x1, x2, . . . , x6 and their velocities are v1, v2, . . . , v6, respectively. This information can be represented by the vector
v=(x1, x2, x3, x4, x5, x6, v1, v2, v3, v4, v5, v6, t) inR13. This vector is called thestateof the particle system at timet.
Operations on Vectors in Rn Our next goal is to define useful operations on vectors inRn. These operations will all be natural extensions of the familiar operations on vectors inR2andR3. We will denote a vectorvinRnusing the notation
v=(v1, v2, . . . , vn) and we will call0=(0,0, . . . ,0)thezero vector.
We noted earlier that inR2andR3two vectors are equivalent (equal) if and only if their corresponding components are the same. Thus, we make the following definition.
DEFINITION2 Vectorsv=(v1, v2, . . . , vn)andw=(w1, w2, . . . , wn)inRnare said to beequivalent(also calledequal) if
v1=w1, v2=w2, . . . , vn=wn
We indicate this by writingv=w.
E X A M P L E 2 Equality of Vectors
(a, b, c, d)=(1,−4,2,7) if and only ifa =1, b= −4, c=2, andd=7.
Our next objective is to define the operations of addition, subtraction, and scalar multiplication for vectors inRn. To motivate these ideas, we will consider how these op- erations can be performed on vectors inR2using components. By studying Figure 3.1.13 you should be able to deduce that ifv=(v1, v2)andw=(w1, w2), then
v+w=(v1+w1, v2+w2) (6)
kv=(kv1, kv2) (7)
In particular, it follows from (7) that
−v=(−1)v=(−v1,−v2) (8)
Figure 3.1.13
y
x v
v + w w
(v1, v2)
v1
(w1, w2)
(v1 + w1, v2 + w2)
w1
w2
v2
y
v x
kv (kv1, kv2) (v1, v2)
v1
kv1
v2
kv2
and hence that
w−v=w+(−v)=(w1−v1, w2−v2) (9) Motivated by Formulas (6)–(9), we make the following definition.
DEFINITION3 Ifv=(v1, v2, . . . , vn)andw=(w1, w2, . . . , wn)are vectors inRn, and ifkis any scalar, then we define
v+w=(v1+w1, v2+w2, . . . , vn+wn) (10)
kv=(kv1, kv2, . . . , kvn) (11)
−v=(−v1,−v2, . . . ,−vn) (12)
w−v=w+(−v)=(w1−v1, w2−v2, . . . , wn−vn) (13)
E X A M P L E 3 Algebraic Operations Using Components Ifv=(1,−3,2)andw=(4,2,1), then
In words, vectors are added (or subtracted) by adding (or sub- tracting) their corresponding components, and a vector is multiplied by a scalar by multi- plying each component by that scalar.
v+w=(5,−1,3), 2v=(2,−6,4)
−w=(−4,−2,−1), v−w=v+(−w)=(−3,−5,1)
The following theorem summarizes the most important properties of vector opera- tions.
THEOREM3.1.1 If u,v,andware vectors inRn,and ifkandmare scalars,then:
(a) u+v=v+u
(b) (u+v)+w=u+(v+w) (c) u+0=0+u=u (d) u+(−u)=0 (e) k(u+v)=ku+kv (f) (k+m)u=ku+mu (g) k(mu)=(km)u (h) 1u=u
We will prove part (b) and leave some of the other proofs as exercises.
Proof (b) Let u=(u1, u2, . . . , un),v=(v1, v2, . . . , vn), and w=(w1, w2, . . . , wn).
Then (u+v)+w=
(u1, u2, . . . , un)+(v1, v2, . . . , vn)
+(w1, w2, . . . , wn)
=(u1+v1, u2+v2, . . . , un+vn)+(w1, w2, . . . , wn) [Vector addition]
=
(u1+v1)+w1, (u2+v2)+w2, . . . , (un+vn)+wn
[Vector addition]
=
u1+(v1+w1), u2+(v2+w2), . . . , un+(vn+wn)
[Regroup]
=(u1, u2, . . . , un)+(v1+w1, v2+w2, . . . , vn+wn) [Vector addition]
=u+(v+w)
The following additional properties of vectors inRncan be deduced easily by ex- pressing the vectors in terms of components (verify).
3.1 Vectors in 2-Space, 3-Space, andn-Space 139
THEOREM3.1.2 Ifvis a vector inRnandkis a scalar,then:
(a) 0v=0 (b) k0=0 (c) (−1)v= −v
Calculating Without Components
One of the powerful consequences of Theorems 3.1.1 and 3.1.2 is that they allow cal- culations to be performed without expressing the vectors in terms of components. For example, suppose thatx,a, andbare vectors inRn, and we want to solve the vector equationx+a=bfor the vectorxwithout using components. We could proceed as follows:
x+a=b [ Given ]
(x+a)+(−a)=b+(−a) [ Add the negative of a to both sides ]
x+(a+(−a))=b−a [ Part (b) of Theorem 3.1.1 ]
x+0=b−a [ Part (d) of Theorem 3.1.1 ]
x=b−a [ Part (c) of Theorem 3.1.1 ]
While this method is obviously more cumbersome than computing with components in Rn, it will become important later in the text where we will encounter more general kinds of vectors.
Linear Combinations Addition, subtraction, and scalar multiplication are frequently used in combination to form new vectors. For example, ifv1,v2, andv3are vectors inRn, then the vectors
u=2v1+3v2+v3 and w=7v1−6v2+8v3 are formed in this way. In general, we make the following definition.
Note that this definition of a linear combination is consis- tent with that given in the con- text of matrices (see Definition 6 in Section 1.3).
DEFINITION4 Ifwis a vector inRn, thenwis said to be alinear combinationof the vectorsv1,v2, . . . ,vrinRnif it can be expressed in the form
w=k1v1+k2v2+ · · · +krvr (14) wherek1, k2, . . . , kr are scalars. These scalars are called thecoefficientsof the linear combination. In the case wherer=1, Formula (14) becomesw=k1v1, so that a linear combination of a single vector is just a scalar multiple of that vector.
Alternative Notations for Vectors
Up to now we have been writing vectors inRnusing the notation
v=(v1, v2, . . . , vn) (15) We call this thecomma-delimitedform. However, since a vector inRnis just a list of itsncomponents in a specific order, any notation that displays those components in the correct order is a valid way of representing the vector. For example, the vector in (15) can be written as
v= [v1 v2 · · · vn] (16) which is calledrow-vectorform, or as
v=
⎡
⎢⎢
⎢⎢
⎣ v1
v2
... vn
⎤
⎥⎥
⎥⎥
⎦ (17)
which is calledcolumn-vectorform. The choice of notation is often a matter of taste or convenience, but sometimes the nature of a problem will suggest a preferred notation.
Notations (15), (16), and (17) will all be used at various places in this text.
Application of Linear Combinations to Color Models Colors on computer monitors are commonly based on what is called theRGBcolor model. Colors in this system are created by adding together percentages of the primary colors red (R), green (G), and blue (B). One way to do this is to identify the primary colors with the vectors
r=(1,0,0) (pure red), g=(0,1,0) (pure green), b=(0,0,1) (pure blue)
inR3and to create all other colors by forming linear combinations ofr,g, andbusing coefficients between 0 and 1, inclusive; these coefficients represent the percentage of each pure color in the mix.
The set of all such color vectors is calledRGBspaceor theRGB color cube(Figure 3.1.14). Thus, each color vectorcin this cube is expressible as a linear combination of the form
c=k1r+k2g+k3b
=k1(1,0,0)+k2(0,1,0)+k3(0,0,1)
=(k1, k2, k3)
where 0≤ki≤1. As indicated in the figure, the corners of the cube represent the pure primary colors together with the colors black, white, magenta, cyan, and yellow. The vectors along the diagonal running from black to white correspond to shades of gray.
Figure 3.1.14
Blue (0, 0, 1)
Cyan (0, 1, 1)
Yellow (1, 1, 0) Red
(1, 0, 0) Magenta (1, 0, 1) Black (0, 0, 0)
Green (0, 1, 0) White (1, 1, 1)
Exercise Set 3.1
In Exercises1–2, find the components of the vector.
1.
y z
x y
(1, 5)
(4, 1) (a)
(0, 0, 4)
(2, 3, 0) (b)
x
2.
x (2, 3) (–3, 3)
(a) (0, 4, 4)
(3, 0, 4) y (b)
y z
x
In Exercises3–4, find the components of the vector−−→
P1P2. 3.(a)P1(3,5), P2(2,8) (b)P1(5,−2,1), P2(2,4,2) 4.(a)P1(−6,2), P2(−4,−1) (b)P1(0,0,0), P2(−1,6,1) 5.(a) Find the terminal point of the vector that is equivalent to
u=(1,2)and whose initial point isA(1,1).
(b) Find the initial point of the vector that is equivalent to u=(1,1,3)and whose terminal point isB(−1,−1,2).
6.(a) Find the initial point of the vector that is equivalent to u=(1,2)and whose terminal point isB(2,0).
(b) Find the terminal point of the vector that is equivalent to u=(1,1,3)and whose initial point isA(0,2,0).
7.Find an initial pointP of a nonzero vectoru=−→
PQwith ter- minal pointQ(3,0,−5)and such that
(a) uhas the same direction asv=(4,−2,−1). (b)uis oppositely directed tov=(4,−2,−1).