This document discusses three types of vectors: numeric vectors, geometric/physical vectors, and functions. Numeric vectors are lists of numbers. Geometric/physical vectors have magnitude and direction, like directed line segments representing displacements. Functions can also be viewed as vectors. All three types of vectors can be added, subtracted, and multiplied by numbers. Numeric vectors correspond to geometric vectors through their components in a coordinate system. Forces are represented as geometric vectors with magnitude and direction.

1. 1.1 - 1
1. Vectors and Vector Operations
1.1 Types of vectors
Three types of vectors that we will be concerned with are the following.
1. numeric vectors – lists of numbers
2. geometric / physical vectors – things with magnitude & direction, e.g. directed
line segments, displacements and forces
3. Functions, e.g y = x2
.
The connection between these different types of vectors is that for each type one can add
and subtract objects of that type and multiply them by numbers. The operations of
addition, subtraction and multiplication by numbers have the same algebraic properties in
each case. Let’s take a look at these three different types of vectors.
1.1.1. Numeric vectors
In this context
a vector is a list of numbers.
For example,
4
-1
7
is a vector. Often the numbers in the list are related in some fashion;
we shall see examples of this as we go along.
The vector
4
-1
7
has three components. The first component is 4, the second component is
-1 and the third component is 7. A vector can have any number of components. For
example,
4
-1
7
0
-8
is a vector with five components. Usually the order of the numbers in the
list is important. For example,
7
-1
4
is a different vector than
4
-1
7
.
When working with vectors, a number is sometimes called a scalar.

2. 1.1 - 2
The components of a vector can be variables or formulas that represent numbers. For
example, we would regard
x
y
z
and
x2
+ 3
-7
cos(y)
as vectors with the understanding that we get
an actual list of numbers when we give numerical values to x, y and z.
We often use a letter to denote a certain vector. For example, we might use the letter x to
denote the vector
4
-1
7
. We indicate this by writing x =
4
-1
7
. When we use a letter to
denote a vector, then a subscript on the letter indicates the particular component of the
vector. For example, x2 indicates the second component of x. For example, if x =
4
-1
7
then x2 = -1.
When we write the components of a vector in a column as with
4
-1
7
, it is called a column
vector. If we write the components of a vector in a row as with (4, -1, 7), it is called a
row vector. Often it doesn’t matter if we write a vector as a column vector or a row
vector, but as we go along we shall see some formulas where it makes a difference. If we
are in a situation where we are distinguishing between column vectors and row vectors
and we want to indicate that we are converting a certain column vector to a row vector or
vice-versa, then we do this with the transpose operation. If x is a certain vector then
xT
= transpose of x
= the same list of numbers as x, but written in row form if x is a
column vector or written in column form if x is a row vector.
For example,
4
-1
7
T
= (4, -1, 7) and (4, -1, 7)T
=
4
-1
7
Example 1.1.1. An electronics company makes two types of circuit boards for computers,
namely ethernet cards and sound cards. Each of these boards requires a certain number of
resistors, capacitors and transistors as follows
ethernet cards sound cards
resistors 5 7
capacitors 2 3
transistors 3 5
There are a number of different vectors that might be of interest in a situation such as this. Here
are some examples.
u =
5
2
3
= vector containing the number of resistors, capacitors and transistors in an ethernet
card

3. 1.1 - 3
v = (5, 7)
= vector containing the number of resistors in an ethernet card and in a sound card
y =
20
30
= vector containing the number of ethernet and sound cards the company plans to make
this coming week
r =
310
130
210
= vector containing the number of resistors, capacitors and transistors the company will
need in order to make 20 ethernet cards and 30 sound cards this coming week week
p = (2, 3, 5)
= vector containing the prices the company has to pay (in cents) for resistors, capacitors
and transistors
1.1.2. Geometric / physical vectors
If you ask a physicist what a vector is he will tell you
A vector is something with magnitude and direction
A good example of a vector of this type is a directed line segment. To construct a
directed line segment we pick two points P and Q and draw the line
segment from P to Q. We put an arrow at the end of the line
segment at Q. We represent the directed line segment from P to Q
by PQ or even just by PQ.
Note that the directed line segment from P to Q is a different directed line segment from the
directed line segment from Q to P, i.e. PQ QP .
In physics one uses directed line segments to describe displacements. Suppose we have a moving
object and we are studying its motion. We fix two times, a starting time s and an ending time t. If
the object is at point P at time s and at point Q at time t, then its net motion over the time interval
from s to t can be described by the directed line segment PQ . This is called the object’s
displacement over the time interval from s to t.
Often we regard two directed line segments as representing the same vector if they have
the same length and direction. Suppose the directed line
segments PQ and RS have the same length and direction. We
might write PQ = RS if we are in a situation where directed
line segments with the same length and direction represent the
same vector. This can be confusing since the actual directed
line segments PQ and RS are different.
P
Q
P
Q
R
S

4. 1.1 - 4
In order to make a connection between directed line segment vectors and numeric vectors,
we draw a coordinate system. If the directed line segments we are working with are all in
the same plane, then we draw a two dimensional xy-coordinate system for that plane. If
they don’t lie in the same plane, we draw a three dimensional xyz-coordinate system for
space. For simplicity, let’s suppose for the moment that they all lie in the same plane.
Suppose the coordinates of P are
x1
y1
.
We shall indicate this by writing P =
x1
y1
.
Suppose also that Q =
x2
y2
. Then to the
directed line segment vector PQ
corresponds the numeric vector
x2 - x1
y2 - y1
.
We shall indicate this by writing
PQ =
x2 - x1
y2 - y1
. Note that x = x2 - x1 is
the change in the x coordinate as we move
from P to Q and y = y2 - y1 is the change
in the y coordinate as we move from P to
Q, so that PQ =
x
y
.
Example 1.1.2. Suppose P =
2
1 and
Q =
5
2 . Then PQ =
x2 - x1
y2 - y1
=
5 - 2
2 - 1 =
3
1 .
Suppose PQ and RS are directed line
segments and PQ = RS , i.e. PQ and
RS have the same length and direction.
Suppose we draw a coordinate system
and PQ =
x2 - x1
y2 - y1
and
RS =
x4 - x3
y4 - y3
. Then the triangles
PQN and RST are congruent. So
x2 - x1 = x4 – x3 and y2 - y1 = y4 – y3. So
PQ and RS are both assigned the
same numeric vector. So directed line
segments that represent the same
P = ( )x1
y1
x
y
x1
x = x2 – x1
x2
y1
y2
PQ = ( )x2 - x1
y2 - y1
= ( )x
y
Q = ( )x2
y2
y = y2 – y1
P = ( )2
1
x
y
1
x = 3
PQ = ( )5 - 2
2 - 1 = ( )3
1
Q = ( )5
2
y = 1
2
1
2
3
3
4
4 5
x
y
x1 x2
y1
y2
P = ( )x1
y1
PQ = ( )x2 - x1
y2 - y1
Q = ( )x2
y2
R = ( )x3
y3
RS = ( )x4 - x3
y4 - y3 S = ( )x4
y4

5. 1.1 - 5
geometric vector correspond to the same numeric vector.
Forces. Forces are another example of a type of physical quantity that is a vector. If you push or
pull on an object with a certain force, then the force F has a certain magnitude, f, and direction, .
For example, suppose we pull on an object with a force of magnitude f = 40 lbs and we pull
making an angle = 30 with the horizontal. Then we can represent this force by a directed line
segment with length equal to 30 and making an angle = 30 with the positive x-axis. If we
choose the starting point of the directed line segment equal to the origin, O, and the ending point
equal to P, then the coordinates of P are
P =
x1
y1
=
f cos( )
f sin( )
. So we can associate to
F the numeric vector
f cos( )
f sin( )
. Often we just
write F =
f cos( )
f sin( )
. For example, if f = 40
and = 30 , then F =
40 cos(30 )
40 sin(30 )
=
40 3/2
40(1/2)
=
20 3
20
=
20 3
20
=
34.64
20 .
1.1.3. Functions
In more advanced work functions are often regarded as vectors. This includes functions
y = f(x) of a single variable, e.g. y = f(x) = x2
. It also includes functions of several
variables w = f(x, y, z) such as w = f(x, y, z) = xey
+ sin(z) and other types of functions. In
the next section we shall see what functions have in common with numeric vectors.
P = ( )f cos
f sin
= ( )40 cos 30
40 sin 30
( )34.7
20
= 30
x
y
10
f cos 34.7
f = 10
20
10
20
30 40
f sin = 20