In this section, we introduce vectors and consider two operations on vectors: scalar multiplication and addition. Let Rdenote the set of all real numbers (that is, all coordinate values on the real number line).
Definition of a Vector
aaDefinition Arealn-vectoris an ordered sequence ofnreal numbers (sometimes referred to as an orderedn-tupleof real numbers). The set of all n-vectors is denotedRn.
For example,R2 is the set of all 2-vectors (ordered 2-tuplesordered pairs) of real numbers; it includes [2,4] and [6.2, 3.14]. R3 is the set of all 3-vectors (ordered 3-tuplesordered triples) of real numbers; it includes [2,3, 0] and [√
2, 42.7,].1
The vector inRn that has allnentries equal to zero is called thezeron-vector.
InR2andR3, the zero vectors are[0, 0]and[0, 0, 0], respectively.
Two vectors inRnareequal if and only if all corresponding entries (calledcoor- dinates) in theirn-tuples agree. That is,[x1,x2,...,xn][y1,y2,...,yn]if and only ifx1y1,x2y2,..., andxnyn.
A single number (such as10 or 2.6) is often called ascalarto distinguish it from a vector.
Geometric Interpretation of Vectors
Vectors inR2frequently represent movement from one point to another in a coordinate plane. From initial point(3, 2)to terminal point(1, 5), there is a net decrease of 2 units along thex-axis and a net increase of 3 units along they-axis. A vector representing this change would thus be[2, 3], as indicated by the arrow in Figure 1.1.
Vectors can be positioned at any desired starting point. For example,[2, 3]could also represent a movement from initial point(9,6)to terminal point(7,3).2
Vectors inR3have a similar geometric interpretation: a 3-vector is used to repre- sent movement between points in three-dimensional space. For example,[2,2, 6]can represent movement from initial point(2, 3,1)to terminal point(4, 1, 5), as shown in Figure 1.2.
1 Many texts distinguish betweenrowvectors, such as[2,3], andcolumnvectors, such as
2
3
. However, in this text, we express vectors as row or column vectors as the situation warrants.
2We use italicized capital letters and parentheses for the points of a coordinate system,such asA(3, 2), and boldface lowercase letters and brackets for vectors, such asx[3, 2].
2 3 4 5 6
(1, 5)
(3, 2) Vector [22, 3]
1
1 2 3 4 5 6 7
x y
21 21 22 23 24 25 26 27
FIGURE 1.1
Movement represented by the vector[2, 3]
1 2 3 4 5
y
x
21 21 22
22232425 23
24 25
2 (4, 1, 5) 3
[2,22, 6]
(2, 3, 21) 4
5
1 1 2 4 3 5
z
21 22 23 24 25
FIGURE 1.2
The vector[2,2, 6]with initial point(2, 3,1)
Three-dimensional movements are usually graphed on a two-dimensional page by slanting the x-axis at an angle to create the optical illusion of three mutually perpendicular axes. Movements are determined on such a graph by breaking them down into components parallel to each of the coordinate axes.
Visualizing vectors inR4and higher dimensions is difficult. However,the same alge- braic principles are involved. For example, the vectorx[2, 7,3, 10]can represent
a movement between points (5,6, 2,1) and (7, 1,1, 9) in a four-dimensional coordinate system.
Length of a Vector
Recall thedistance formulain the plane;the distance between two points(x1,y1)and (x2,y2)isd
(x2x1)2(y2y1)2(see Figure 1.3). This formula arises from the Pythagorean Theorem for right triangles. The 2-vector between the points is[a1,a2], where a1x2x1 anda2y2y1, so d
a21a22. This formula motivates the following definition:
Definition Thelength (also known as thenormormagnitude) of a vectora [a1,a2,...,an]inRnisa
a21a22ãããa2n.
Example 1
The length of the vectora[4,3, 0, 2]is given by
a
42(3)20222
1694√ 29.
Note that the length of any vector in Rn is always nonnegative (that is, 0).
(Do you know why this statement is true?) Also, the only vector with length 0 in Rnis the zero vector[0, 0,..., 0](why?).
Vectors of length 1 play an important role in linear algebra.
(x2,y2)
(x1,y1)
x1 x2
y
x Vector
a5[a1,a2]
a25y22y1 a15x22x1
y1 A
B y2
FIGURE 1.3
The line segment (and vector) connecting pointsAandB, with length
(x2x1)2(y2y1)2
a21a22
Definition Any vector of length 1 is called aunit vector.
InR2, the vector 35,45
is a unit vector, because 3
5
2
452
1. Similarly, 0,35, 0,45
is a unit vector inR4. Certain unit vectors are particularly useful: those with a single coordinate equal to 1 and all other coordinates equal to 0. InR2these vectors are denotedi[1, 0]andj[0, 1]; in R3 they are denotedi[1, 0, 0],j [0, 1, 0],andk[0, 0, 1]. InRn,these vectors,thestandard unit vectors,are denoted e1[1, 0, 0,..., 0],e2[0, 1, 0,..., 0],...,en[0, 0, 0,..., 1].
Scalar Multiplication and Parallel Vectors
aaDefinition Letx[x1,x2,...,xn]be a vector inRn, and letcbe any scalar (real number). Thencx, thescalar multiple of x byc, is the vector[cx1,cx2,...,cxn].
For example, if x[4,5], then 2x[8,10],3x[12, 15], and 12x 2,52
. These vectors are graphed in Figure 1.4. From the graph, you can see that
2 2 1
2
4 6 8 10 12 14 16
4 6 8 10 12 14
22 24
22 24 26 26
28 210 212 214
23x
2x x 2
216
210 28 x
FIGURE 1.4
Scalar multiples ofx[4,5](all vectors drawn with initial point at origin)
the vector 2xpoints in the same direction asxbut is twice as long. The vectors3x and12x indicate movements in the direction opposite tox, with3xbeing three times as long asxand12xbeing half as long.
In general, inRn, multiplication byc dilates(expands) the length of the vector when |c|>1 and contracts (shrinks) the length when |c|<1. Scalar multiplica- tion by 1 or 1 does not affect the length. Scalar multiplication by 0 always yields the zero vector. These properties are all special cases of the following theorem:
Theorem 1.1 Let x∈Rn, and let c be any real number (scalar). Then cx
|c| x. That is, the length ofcxis the absolute value ofctimes the length ofx.
The proof of Theorem 1.1 is left as Exercise 23 at the end of this section.
We have noted that in R2, the vector cx is in the same direction as x when c is positive and in the direction opposite to x when c is negative, but have not yet discussed “direction” in higher-dimensional coordinate systems. We use scalar multiplication to give a precise definition for vectors having the same or opposite directions.
Definition Two nonzero vectorsxandy inRnarein the same directionif and only if there is a positive real numbercsuch thatycx. Two nonzero vectorsx andyarein opposite directionsif and only if there is a negative real numberc such thatycx. Two nonzero vectors areparallelif and only if they are either in the same direction or in the opposite direction.
Hence, vectors [1,3, 2] and [3,9, 6] are in the same direction, because [3,9, 6]3[1,3, 2] (or because[1,3, 2]13[3,9, 6]), as shown in Figure 1.5.
Similarly, vectors[3, 6, 0, 15]and[4,8, 0,20]are in opposite directions, because [4,8, 0,20] 43[3, 6, 0, 15].
The next result follows from Theorem 1.1:
Corollary 1.2 Ifxis a nonzero vector inRn, thenu(1/x)xis a unit vector in the same direction asx.
Proof. The vectoruin Corollary 1.2 is certainly in the same direction asxbecauseuis a pos- itive scalar multiple ofx(the scalar is1/x). Also, by Theorem 1.1,u(1/x)x (1/x)x1, souis a unit vector.
This process of “dividing” a vector by its length to obtain a unit vector in the same direction is callednormalizingthe vector (see Figure 1.6).
21
21 21 22
23 24
22 23 24 22
23 24 25 28
29
210 27 26
5 4 3 1
6 x
y z
5 4
1 2 3
1 2 3 4 [3,29, 6]
[1,23, 2]
2
FIGURE 1.5
The parallel vectors[1,3, 2]and[3,9, 6]
x
u x1, ))x))
x2, ))x))
x3 [x1,x2,x3]
))x))
FIGURE 1.6
Normalizing a vectorxto obtain a unit vectoruin the same direction (withx>1)
Example 2
Consider the vector[2, 3,1, 1]inR4. Because[2, 3,1, 1]√
15, normalizing[2, 3,1, 1]
gives a unit vectoruin the same direction as[2, 3,1, 1], which is u
1
√15
[2, 3,1, 1]
2
√15, 3
√15,√1 15, 1
√15
.
Addition and Subtraction with Vectors
aaDefinition Letx[x1,x2,...,xn]andy[y1,y2,...,yn]be vectors inRn. Then xy, thesumofxandy, is the vector[x1y1,x2y2,...,xnyn]inRn.
Vectors are added by summing their respective coordinates. For exam- ple, if x[2,3, 5] and y[6, 4,2], then xy[26,34, 52] [4, 1, 3]. Vectors cannot be added unless they have the same number of coordinates.
There is a natural geometric interpretation for the sum of vectors in a plane or in space. Draw a vectorx. Then draw a vectoryfrom the terminal point ofx. The sum of xandyis the vector whoseinitialpoint is the same as that ofxand whoseterminal point is the same as that ofy. The total movement(xy)is equivalent to first moving alongxand then alongy. Figure 1.7 illustrates this inR2.
Let y denote the scalar multiple 1y. We can now define subtraction of vectors in a natural way: if x and y are both vectors in Rn, let xy be the vector x(y). A geometric interpretation of this is in Figure 1.8 (move- ment x followed by movement y). An alternative interpretation is described in Exercise 11.
Fundamental Properties of Addition and Scalar Multiplication
Theorem 1.3 contains the basic properties of addition and scalar multiplication of vectors. Thecommutative,associative, anddistributivelaws are so named because they resemble the corresponding laws for real numbers.
x y
x2y2
xⴙy
x[x1,x2] y[y1,y2] x1y1 y2
y y
x
x x2
y1 x1
FIGURE 1.7
Addition of vectors inR2
x2y x
x y
2y 2y
FIGURE 1.8
Subtraction of vectors inR2:xyx(y)
Theorem 1.3 Let x[x1,x2,...,xn],y[y1,y2,...,yn], and z[z1,z2,...,zn]be any vectors inRn, and letcanddbe any real numbers (scalars). Let0represent the zero vector inRn. Then
(1) xyyx Commutative Law of Addition (2) x(yz)(xy)z Associative Law of Addition
(3) 0xx0x Existence of Identity Element for Addition (4) x(x)(x)x0 Existence of Inverse Elements for Addition (5) c(xy)cxcy Distributive Laws of Scalar Multiplication (6) (cd)xcxdx over Addition
(7) (cd)xc(dx) Associativity of Scalar Multiplication (8) 1xx Identity Property for Scalar Multiplication
In part (3), the vector0is called anidentity elementfor addition because0does not change the identity of any vector to which it is added. A similar statement is true in part (8) for the scalar 1 with scalar multiplication. In part (4), the vectorxis called theadditive inverse element of xbecause it “cancels outx” to produce the zero vector.
Each part of the theorem is proved by calculating the entries in each coordinate of the vectors and applying a corresponding law for real-number arithmetic. We illustrate thiscoordinate-wisetechnique by proving part (6). You are asked to prove other parts of the theorem in Exercise 24.
Proof. Proof of Part (6):
(cd)x(cd)[x1,x2,...,xn]
[(cd)x1,(cd)x2,...,(cd)xn] definition of scalar multiplication
[cx1dx1,cx2dx2,...,cxndxn] coordinate-wise use of distributive law inR [cx1,cx2,...,cxn][dx1,dx2,...,dxn] definition of vector addition
c[x1,x2,...,xn]d[x1,x2,...,xn] definition of scalar multiplication cxdx.
The following theorem is very useful (the proof is left as Exercise 25):
Theorem 1.4 Letxbe a vector inRn, and letcbe a scalar. Ifcx0, then eitherc0 orx0.
Linear Combinations of Vectors
aa
Definition Letv1,v2,...,vkbe vectors inRn. Then the vectorvis alinear com- bination ofv1,v2,...,vk if and only if there are scalars c1,c2,...,ck such that vc1v1c2v2ãããckvk.
Thus, a linear combination of vectors is a sum of scalar multiples of those vectors.
For example, the vector[2, 8, 5, 0]is a linear combination of[3, 1,2, 2],[1, 0, 3,1], and[4,2, 1, 0]because 2[3, 1,2, 2]4[1, 0, 3,1]3[4,2, 1, 0][2, 8, 5, 0].
Note that any vector in R3 can be expressed in a unique way as a linear com- bination of i,j, and k. For example, [3,2, 5]3[1, 0, 0]2[0, 1, 0]5[0, 0, 1]
3i2j5k. In general, [a,b,c]aibjck. Also, every vector in Rn can be expressed as a linear combination of the standard unit vectorse1[1, 0, 0,..., 0],e2 [0, 1, 0,..., 0],...,en[0, 0,..., 0, 1](why?).
One helpful way to picture linear combinations of the vectorsv1,v2,...,vk is to remember that each vector represents a certain amount of movement in a particular direction. When we combine these vectors using addition and scalar multiplication, the endpoint of each linear combination vector represents a “destination” that can be reached using these operations. For example, the linear combination w2[1, 3]
1
2[4,5]3[2,1][6,112]is the destination reached by traveling in the direction of [1, 3], but traveling twice its length, then traveling in the direction opposite to[4,5], but half its length, and finally traveling in the direction [2,1], but three times its length (see Figure 1.9(a)).
We can also consider the set of all possible destinations that can be reached using linear combinations of a certain set of vectors. For example, the set of all linear com- binations inR3ofv1[2, 0, 1]andv2[0, 1,2]is the set of all vectors (beginning at the origin) with endpoints lying in the plane through the origin containingv1and v2(see Figure 1.9(b)).
Physical Applications of Addition and Scalar Multiplication
Addition and scalar multiplication of vectors are often used to solve problems in ele- mentary physics. Recall the trigonometric fact that ifvis a vector inR2forming an angle ofwith the positivex-axis, thenv[vcos,vsin], as in Figure 1.10.
(a) (b) 1
2[4,5]
[4,5]
3[2,1]
[2,1]
2[1,3]
[1,3] w
[2,0, 1]
[0,1,2]
FIGURE 1.9
(a) The destinationw2[1, 3]12[4,5]3[2,1] 6,112
; (b) the plane inR3containing all linear combinations of[2, 0, 1]and[0, 1,2]
))v))sin
))v))cos
v y
x
FIGURE 1.10
The vectorv[vcos,vsin]forming an angle ofwith the positivex-axis
Example 3
Resultant Velocity:Suppose a man swims5 km/hrin calm water. If he is swimming toward the east in a wide stream with a northwest current of3 km/hr, what is hisresultant velocity(net speed and direction)?
The velocities of the swimmer and current are shown as vectors in Figure 1.11, where we have, for convenience, placed the swimmer at the origin. Now, v1[5, 0] and v2 [3cos135◦, 3sin135◦] 3√
2/2, 3√ 2/2
. Thus, the total (resultant) velocity of the swimmer is the sum of these velocities,v1v2, which is 53√
2/2, 3√ 2/2
≈ [2.88, 2.12]. Hence, each
v1⫹v2 v2
v1 3 2
2
y North ,
Swimmer
South
East [5, 0]
West
Current
Resultant
x 3 2
2 2 3 22 ,3 2
52 2
FIGURE 1.11
Velocityv1of swimmer, velocityv2of current, and resultant velocityv1v2
hour the swimmer is traveling about2.9 kmeast and2.1 kmnorth. The resultant speed of the swimmer is 53√
2/2, 3√
2/2≈3.58 km/hr.
Example 4
Newton’s Second Law:Newton’s famousSecond Law of Motionasserts that the sum,f, of the vector forces on an object is equal to the scalar multiple of the mass mof the object times the vector accelerationaof the object; that is,fma. For example, suppose a mass of5 kg (kilograms) in a three-dimensional coordinate system has two forces acting on it: a forcef1of 10 newtons3in the direction of the vector[2, 1, 2]and a forcef2of20newtons in the direction of the vector[6, 3,2]. What is the acceleration of the object?
We must first normalize the direction vectors [2, 1, 2] and [6, 3,2] so that their lengths do not contribute to the magnitude of the forces f1 and f2. Therefore, f1 10([2, 1, 2]/[2, 1, 2]), andf220([6, 3,2]/[6, 3,2]). The net force on the object is ff1f2. Thus, the net acceleration on the object is
a 1 mf 1
m(f1f2)1 5
10
[2, 1, 2]
[2, 1, 2]
20
[6, 3,2]
[6, 3,2]
,
which equals 23[2, 1, 2]47[6, 3,2]
44 21,5021,214
. The length ofais approximately3.18, so pulling out a factor of3.18from each coordinate, we can approximateaas3.18[0.66, 0.75, 0.06], where [0.66, 0.75, 0.06] is aunit vector. Hence, the acceleration is about 3.18 m/sec2 in the direction[0.66, 0.75, 0.06].
31 newton1 kg-m/sec2
kilogram-meter/second2
,or the force needed to push 1 kg at a speed 1 m/sec (meter per second) faster every second.
If the sum of the forces on an object is0, then the object is inequilibrium; there is no acceleration in any direction (see Exercise 21).
New Vocabulary
addition of vectors additive inverse vector
associative law for scalar multiplication associative law for vector addition commutative law for vector addition contraction of a vector
dilation of a vector distance formula
distributive laws for vectors equilibrium
initial point of a vector
length (norm, magnitude) of a vector linear combination of vectors
normalization of a vector opposite direction vectors parallel vectors
realn-vector resultant velocity same direction vectors scalar
scalar multiplication of a vector standard unit vectors
subtraction of vectors terminal point of a vector unit vector
zeron-vector
Highlights
■ n-vectors are used to represent movement from one point to another in an n-dimensional coordinate system.
■ The norm (length) of a vector is the distance from its intitial point to its terminal point and is nonnegative.
■ Multiplication of a nonzero vector by a nonzero scalar results in a vector that is parallel to the original.
■ For any given nonzero vector,there is auniqueunit vector in the same direction.
■ The sum and difference of two vectors inR2can be found using the diagonals of appropriate parallelograms.
■ The commutative, associative, and distributive laws hold for addition of vectors inRn.
■ If the scalar multiple of a vector is the zero vector, then either the scalar is zero or the vector is the zero vector.
■ Every vector inRnis a linear combination of the standard unit vectors inRn.
■ The linear combinations of a given set of vectors represent the set of all possible
“destinations” that can be reached using those vectors.
■ Any vectorvinR2can be expressed as[||v||cos,||v||sin], whereis the angle vforms with the positivex-axis.
■The resultant velocity of an object is the sum of its individual vector velocities.
■The sum of the vector forces on an object is equal to the scalar product of the object’s mass and its acceleration vector.