CHAPTER

C H1A P T E R 1

n n
Vectors in R and C ,
Spatial Vectors
1.1 Introduction
There are two ways to motivate the notion of a vector: one is by means of lists of numbers and subscripts,
and the other is by means of certain objects in physics. We discuss these two ways below.
Here we assume the reader is familiar with the elementary properties of the field of real numbers,
denoted by R. On the other hand, we will review properties of the field of complex numbers, denoted by
C. In the context of vectors, the elements of our number fields are called scalars.
Although we will restrict ourselves in this chapter to vectors whose elements come from R and then
from C, many of our operations also apply to vectors whose entries come from some arbitrary field K.

Lists of Numbers
Suppose the weights (in pounds) of eight students are listed as follows:
156; 125; 145; 134; 178; 145; 162; 193
One can denote all the values in the list using only one symbol, say w, but with different subscripts; that is,
w1 ; w2 ; w3 ; w4 ; w5 ; w6 ; w7 ; w8
Observe that each subscript denotes the position of the value in the list. For example,
w1 ¼ 156; the first number; w2 ¼ 125; the second number; . . .
Such a list of values,
w ¼ ðw1 ; w2 ; w3 ; . . . ; w8 Þ
is called a linear array or vector.

Vectors in Physics
Many physical quantities, such as temperature and speed, possess only ‘‘magnitude.’’ These quantities
can be represented by real numbers and are called scalars. On the other hand, there are also quantities,
such as force and velocity, that possess both ‘‘magnitude’’ and ‘‘direction.’’ These quantities, which can
be represented by arrows having appropriate lengths and directions and emanating from some given
reference point O, are called vectors.
Now we assume the reader is familiar with the space R3 where all the points in space are represented
by ordered triples of real numbers. Suppose the origin of the axes in R3 is chosen as the reference point O
for the vectors discussed above. Then every vector is uniquely determined by the coordinates of its
endpoint, and vice versa.
There are two important operations, vector addition and scalar multiplication, associated with vectors
in physics. The definition of these operations and the relationship between these operations and the
endpoints of the vectors are as follows.

1
 2 CHAPTER 1 Vectors in Rn and Cn, Spatial Vectors

Figure 1-1

(i) Vector Addition: The resultant u þ v of two vectors u and v is obtained by the parallelogram law;
that is, u þ v is the diagonal of the parallelogram formed by u and v. Furthermore, if ða; b; cÞ and
ða0 ; b0 ; c0 Þ are the endpoints of the vectors u and v, then ða þ a0 ; b þ b0 ; c þ c0 Þ is the endpoint of the
vector u þ v. These properties are pictured in Fig. 1-1(a).
(ii) Scalar Multiplication: The product ku of a vector u by a real number k is obtained by multiplying
the magnitude of u by k and retaining the same direction if k > 0 or the opposite direction if k < 0.
Also, if ða; b; cÞ is the endpoint of the vector u, then ðka; kb; kcÞ is the endpoint of the vector ku.
These properties are pictured in Fig. 1-1(b).
Mathematically, we identify the vector u with its ða; b; cÞ and write u ¼ ða; b; cÞ. Moreover, we call
the ordered triple ða; b; cÞ of real numbers a point or vector depending upon its interpretation. We
generalize this notion and call an n-tuple ða1 ; a2 ; . . . ; an Þ of real numbers a vector. However, special
notation may be used for the vectors in R3 called spatial vectors (Section 1.6).

1.2 Vectors in Rn
The set of all n-tuples of real numbers, denoted by Rn , is called n-space. A particular n-tuple in Rn , say

u ¼ ða1 ; a2 ; . . . ; an Þ

is called a point or vector. The numbers ai are called the coordinates, components, entries, or elements
of u. Moreover, when discussing the space Rn , we use the term scalar for the elements of R.
Two vectors, u and v, are equal, written u ¼ v, if they have the same number of components and if the
corresponding components are equal. Although the vectors ð1; 2; 3Þ and ð2; 3; 1Þ contain the same three
numbers, these vectors are not equal because corresponding entries are not equal.
The vector ð0; 0; . . . ; 0Þ whose entries are all 0 is called the zero vector and is usually denoted by 0.

EXAMPLE 1.1
(a) The following are vectors:

ð2; 5Þ; ð7; 9Þ; ð0; 0; 0Þ; ð3; 4; 5Þ

The first two vectors belong to R2 , whereas the last two belong to R3 . The third is the zero vector in R3 .
(b) Find x; y; z such that ðx  y; x þ y; z  1Þ ¼ ð4; 2; 3Þ.
By definition of equality of vectors, corresponding entries must be equal. Thus,

x  y ¼ 4; x þ y ¼ 2; z1¼3

Solving the above system of equations yields x ¼ 3, y ¼ 1, z ¼ 4.

CHAPTER 1 Vectors in Rn and Cn, Spatial Vectors 3

Column Vectors
Sometimes a vector in n-space Rn is written vertically rather than horizontally. Such a vector is called a
column vector, and, in this context, the horizontally written vectors in Example 1.1 are called row
vectors. For example, the following are column vectors with 2; 2; 3, and 3 components, respectively:
2 3 2 1:5 3
    1
1 3 6 27
; ; 4 5 5; 4 35
2 4
6 15

We also note that any operation defined for row vectors is defined analogously for column vectors.

1.3 Vector Addition and Scalar Multiplication

Consider two vectors u and v in Rn , say
u ¼ ða1 ; a2 ; . . . ; an Þ and v ¼ ðb1 ; b2 ; . . . ; bn Þ

Their sum, written u þ v, is the vector obtained by adding corresponding components from u and v. That is,
u þ v ¼ ða1 þ b1 ; a2 þ b2 ; . . . ; an þ bn Þ

The scalar product or, simply, product, of the vector u by a real number k, written ku, is the vector
obtained by multiplying each component of u by k. That is,
ku ¼ kða1 ; a2 ; . . . ; an Þ ¼ ðka1 ; ka2 ; . . . ; kan Þ

Observe that u þ v and ku are also vectors in Rn . The sum of vectors with different numbers of
components is not defined.
Negatives and subtraction are defined in Rn as follows:

u ¼ ð1Þu and u  v ¼ u þ ðvÞ

The vector u is called the negative of u, and u  v is called the difference of u and v.
Now suppose we are given vectors u1 ; u2 ; . . . ; um in Rn and scalars k1 ; k2 ; . . . ; km in R. We can
multiply the vectors by the corresponding scalars and then add the resultant scalar products to form the
vector
v ¼ k1 u1 þ k2 u2 þ k3 u3 þ    þ km um
Such a vector v is called a linear combination of the vectors u1 ; u2 ; . . . ; um .

EXAMPLE 1.2
(a) Let u ¼ ð2; 4; 5Þ and v ¼ ð1; 6; 9Þ. Then
u þ v ¼ ð2 þ 1; 4 þ ð5Þ; 5 þ 9Þ ¼ ð3; 1; 4Þ
7u ¼ ð7ð2Þ; 7ð4Þ; 7ð5ÞÞ ¼ ð14; 28; 35Þ
v ¼ ð1Þð1; 6; 9Þ ¼ ð1; 6; 9Þ
3u  5v ¼ ð6; 12; 15Þ þ ð5; 30; 45Þ ¼ ð1; 42; 60Þ

(b) The zero vector 0 ¼ ð0; 0; . . . ; 0Þ in Rn is similar to the scalar 0 in that, for any vector u ¼ ða1 ; a2 ; . . . ; an Þ.
u þ 0 ¼ ða1 þ 0; a2 þ 0; . . . ; an þ 0Þ ¼ ða1 ; a2 ; . . . ; an Þ ¼ u
2 3 2 3 2 3 2 3 2 3
2 3 4 9 5
(c) Let u ¼ 4 3 5 and v ¼ 4 1 5. Then 2u  3v ¼ 4 6 5 þ 4 3 5 ¼ 4 9 5.
4 2 8 6 2
 4 CHAPTER 1 Vectors in Rn and Cn, Spatial Vectors

Basic properties of vectors under the operations of vector addition and scalar multiplication are
described in the following theorem.

THEOREM 1.1: For any vectors u; v; w in Rn and any scalars k; k 0 in R,

(i) ðu þ vÞ þ w ¼ u þ ðv þ wÞ, (v) kðu þ vÞ ¼ ku þ kv,
(ii) u þ 0 ¼ u; (vi) ðk þ k 0 Þu ¼ ku þ k 0 u,
(iii) u þ ðuÞ ¼ 0; (vii) (kk’)u=k(k’u);
(iv) u þ v ¼ v þ u, (viii) 1u ¼ u.

We postpone the proof of Theorem 1.1 until Chapter 2, where it appears in the context of matrices
(Problem 2.3).
Suppose u and v are vectors in Rn for which u ¼ kv for some nonzero scalar k in R. Then u is called a
multiple of v. Also, u is said to be in the same or opposite direction as v according to whether k > 0 or
k < 0.

1.4 Dot (Inner) Product

Consider arbitrary vectors u and v in Rn ; say,
u ¼ ða1 ; a2 ; . . . ; an Þ and v ¼ ðb1 ; b2 ; . . . ; bn Þ
The dot product or inner product or scalar product of u and v is denoted and defined by
u  v ¼ a1 b1 þ a2 b2 þ    þ an bn
That is, u  v is obtained by multiplying corresponding components and adding the resulting products.
The vectors u and v are said to be orthogonal (or perpendicular) if their dot product is zero—that is, if
u  v ¼ 0.

EXAMPLE 1.3
(a) Let u ¼ ð1; 2; 3Þ, v ¼ ð4; 5; 1Þ, w ¼ ð2; 7; 4Þ. Then,
u  v ¼ 1ð4Þ  2ð5Þ þ 3ð1Þ ¼ 4  10  3 ¼ 9
u  w ¼ 2  14 þ 12 ¼ 0; v  w ¼ 8 þ 35  4 ¼ 39
Thus, u and w are orthogonal.
2 3 2 3
2 3
(b) Let u ¼ 4 3 5 and v ¼ 4 1 5. Then u  v ¼ 6  3 þ 8 ¼ 11.
4 2
(c) Suppose u ¼ ð1; 2; 3; 4Þ and v ¼ ð6; k; 8; 2Þ. Find k so that u and v are orthogonal.

First obtain u  v ¼ 6 þ 2k  24 þ 8 ¼ 10 þ 2k. Then set u  v ¼ 0 and solve for k:

10 þ 2k ¼ 0 or 2k ¼ 10 or k¼5
Basic properties of the dot product in Rn (proved in Problem 1.13) follow.

THEOREM 1.2: For any vectors u; v; w in Rn and any scalar k in R:

(i) ðu þ vÞ  w ¼ u  w þ v  w; (iii) u  v ¼ v  u,
(ii) ðkuÞ  v ¼ kðu  vÞ, (iv) u  u  0; and u  u ¼ 0 iff u ¼ 0.

Note that (ii) says that we can ‘‘take k out’’ from the first position in an inner product. By (iii) and (ii),
u  ðkvÞ ¼ ðkvÞ  u ¼ kðv  uÞ ¼ kðu  vÞ
CHAPTER 1 Vectors in Rn and Cn, Spatial Vectors 5

That is, we can also ‘‘take k out’’ from the second position in an inner product.
The space Rn with the above operations of vector addition, scalar multiplication, and dot product is
usually called Euclidean n-space.

Norm (Length) of a Vector

The norm or length of a vector u in Rn , denoted by kuk, is defined to be the nonnegative square root of
u  u. In particular, if u ¼ ða1 ; a2 ; . . . ; an Þ, then
pffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
kuk ¼ u  u ¼ a21 þ a22 þ    þ a2n
That is, kuk is the square root of the sum of the squares of the components of u. Thus, kuk  0, and
kuk ¼ 0 if and only if u ¼ 0.
A vector u is called a unit vector if kuk ¼ 1 or, equivalently, if u  u ¼ 1. For any nonzero vector v in
Rn , the vector
1 v
v^ ¼ v¼
kvk kvk
is the unique unit vector in the same direction as v. The process of finding v^ from v is called normalizing v.

EXAMPLE 1.4
(a) Suppose u ¼ ð1; 2; 4; 5; 3Þ. To find kuk, we can first find kuk2 ¼ u  u by squaring each component of u and
adding, as follows:
kuk2 ¼ 12 þ ð2Þ2 þ ð4Þ2 þ 52 þ 32 ¼ 1 þ 4 þ 16 þ 25 þ 9 ¼ 55
pffiffiffiffiffi
Then kuk ¼ 55.
(b) Let v ¼ ð1; 3; 4; 2Þ and w ¼ ð12 ;  16 ; 56 ; 16Þ. Then
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi 9 1 25 1 36 pffiffiffi
kvk ¼ 1 þ 9 þ 16 þ 4 ¼ 30 and kwk ¼ þ þ þ ¼ ¼ 1¼1
36 36 36 36 36
Thus w is a unit vector, but v is not a unit vector. However, we can normalize v as follows:
 
v 1 3 4 2
v^ ¼ ¼ pffiffiffiffiffi ; pffiffiffiffiffi ; pffiffiffiffiffi ; pffiffiffiffiffi
kvk 30 30 30 30
This is the unique unit vector in the same direction as v.

The following formula (proved in Problem 1.14) is known as the Schwarz inequality or Cauchy–
Schwarz inequality. It is used in many branches of mathematics.

THEOREM 1.3 (Schwarz): For any vectors u; v in Rn , ju  vj  kukkvk.

Using the above inequality, we also prove (Problem 1.15) the following result known as the ‘‘triangle
inequality’’ or Minkowski’s inequality.

THEOREM 1.4 (Minkowski): For any vectors u; v in Rn , ku þ vk  kuk þ kvk.

Distance, Angles, Projections

The distance between vectors u ¼ ða1 ; a2 ; . . . ; an Þ and v ¼ ðb1 ; b2 ; . . . ; bn Þ in Rn is denoted and defined
by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dðu; vÞ ¼ ku  vk ¼ ða1  b1 Þ2 þ ða2  b2 Þ2 þ    þ ðan  bn Þ2
One can show that this definition agrees with the usual notion of distance in the Euclidean plane R2 or
space R3 .
 6 CHAPTER 1 Vectors in Rn and Cn, Spatial Vectors

The angle y between nonzero vectors u; v in Rn is defined by

uv
cos y ¼
kukkvk
This definition is well defined, because, by the Schwarz inequality (Theorem 1.3),
uv
1  1
kukkvk
Note that if u  v ¼ 0, then y ¼ 90 (or y ¼ p=2). This then agrees with our previous definition of
orthogonality.
The projection of a vector u onto a nonzero vector v is the vector denoted and defined by
uv uv
projðu; vÞ ¼ 2
v¼ v
kvk vv
We show below that this agrees with the usual notion of vector projection in physics.
EXAMPLE 1.5
(a) Suppose u ¼ ð1; 2; 3Þ and v ¼ ð2; 4; 5Þ. Then
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi
dðu; vÞ ¼ ð1  2Þ2 þ ð2  4Þ2 þ ð3  5Þ2 ¼ 1 þ 36 þ 4 ¼ 41
To find cos y, where y is the angle between u and v, we first find
u  v ¼ 2  8 þ 15 ¼ 9; kuk2 ¼ 1 þ 4 þ 9 ¼ 14; kvk2 ¼ 4 þ 16 þ 25 ¼ 45
Then
uv 9
cos y ¼ ¼ pffiffiffiffiffipffiffiffiffiffi
kukkvk 14 45
Also,
 
uv
9 1 2 4
projðu; vÞ ¼ 2
v ¼ ð2; 4; 5Þ ¼ ð2; 4; 5Þ ¼ ; ;1
kvk 45 5 5 5

(b) Consider the vectors u and v in Fig. 1-2(a) (with respective endpoints A and B). The (perpendicular) projection
of u onto v is the vector u* with magnitude
uv uv
ku*k ¼ kuk cos y ¼ kuk ¼
kukvk kvk
To obtain u*, we multiply its magnitude by the unit vector in the direction of v, obtaining
v uv v uv
u* ¼ ku*k ¼ ¼ v
kvk kvk kvk kvk2
This is the same as the above definition of projðu; vÞ.

A z

P(b1–a1, b2–a2 , b3–a3 )

u B(b1, b2, b3)

u A(a1, a2, a3)

θ u* C
0 B 0
y

x
Projection u* of u onto u=B–A

(a ) (b)

Figure 1-2