# PG Reading 06_vectors pdf (PDF , 76kb) ```Communications Engineering MSc - Preliminary Reading
6 Getting Started With… Vectors
Simple numbers, for example 2, 4, -6, 3.72 or , can be represented by single points
somewhere on the real number line. They represent one thing. They are ideal for representing
what are called scalar quantities (for example the temperature, the distance from London, the
length of a telephone conversation) but they can’t easily represent quantities that have two or
more components.
y-axis
For example, the difference in location between two points in two-dimensional space has two
scalar components – usually measured along orthogonal axes, such as the x-, and y- axes of the
Cartesian (rectangular) co-ordinate system. If I wanted to express this quantity in numbers, I’d
need to use two numbers. In algebra, I’d need to give it two variable names, perhaps writing it
in terms of the two components (the distances along each axis), as a co-ordinate: something
like (x, y). As an example, consider the distance from the origin to the point (4,3):
(4,3)
3
x-axis
4
Figure 1 - The Line from the Origin to (4,3) as a Vector
That’s not too bad if there are only two components, but what if there are ten? Or fifty?
Equations would rapidly get rather tedious and time-consuming to write out. It would be
helpful to find a notation that allowed the entire multi-component quantity to be represented by
just one symbol – it would save a lot of time writing out equations. Enter vectors.
6.1 Vectors
A vector is just that – a way of writing a quantity with multiple components with just one
symbol. They are usually written in bold, e.g. x. (When writing them out by hand with a pen,
it’s not so easy to write in bold, so they are often written with a single underline, e.g. x.)
Probably the most familiar example of a vector is the distance between two points in space. In
the usual three-dimensional space that we live in, this requires a vector with three
components1. Another common example is velocity – anything moving has both a speed and a
1
At least if you’re an engineer it does. Physicists sometimes theorise that there are rather more dimensions, perhaps
eleven, it’s just that the entire observable Universe consists of just one value of eight of them. What would happen
if you could travel some distance along one of the other eight dimensions is not something engineers spend a lot of
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direction, and this can most conveniently be represented in terms of a component of the speed
in these three orthogonal directions2.
This introduces an important feature of vectors – they have a magnitude (or amplitude), which
is independent of their direction. The magnitude in the examples above would be the distance
between the two points along the straight line joining them together, and the speed. And they
also have a direction, sometimes written in terms of a vector with unit length in the same
direction, sometimes called a unit vector.
Three-component vectors (sometimes called three-dimension vectors) can be thought of as a
line in space, starting at one point and ending at another, with the length of the line being the
amplitude of the vector, and the direction of the line being the direction.
(Note that you can’t put any three quantities together and call them a vector – they all have to
be the same sort of quantity, measured in the same units, and they have to be components of
the same overall ‘thing’, preferably along orthogonal directions. Distances to a point along the
x-, y- and z- axes is fine, but a vector can’t have completely unrelated components – for
example the price of gold, the population of India, and the temperature in Tokyo.)
Thinking of vectors as a line (which is often a useful way to think about them, even when they
have more than three components) suggests a way in which we can define addition and
subtraction for vectors. Suppose two vectors represent different distances travelled in different
directions: the addition of the two vectors is just the total distance from the starting point if you
travel the distance and direction indicated by the first vector, followed by the distance and
direction indicated by the second:
c-d
a
d
b
c
a+b
Figure 2 - Adding and Subtracting Vectors
(Subtraction, as you can also see from the diagram, can be readily defined in the same terms –
so that (c – d) + d = c, as you might expect.)
In terms of the components of the vectors, this is the same thing as just adding up the
individual components. For example, consider a vector that represents a distance 2 meters
east, 3 meters north, and 5 meters up. This could be written:
e  2
3
5
And another vector that represents a distance 3 meters west, 2 meters south, and 3 meters
down:
2
For example, moving north-east at 2 metres per second is the same thing as moving north at one metre per
second, and moving east at one metre per second at the same time.
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f   3
3
2
If you travelled the first distance, and then the second distance, you would end up having
travelled a distance:
e  f   1
1
2
which is one meter south, one meter east and two meters above where you started. Vectors are
always added and subtracted like this3.
6.3 Some More Notation
A vector a has an amplitude, often written as |a|, which is the length of the line, and also a
direction. A unit vector (that is, a vector with an amplitude of one) in the direction of a is
written as a . A three-dimensional vector representing a distance can therefore be expressed
as:
a  ax x  a y y  az z
where x is a unit vector in a direction along the x-axis, y a unit vector along the y-axis, and z
a unit vector along the z-axis, and ax, ay and az are the scalar components of this vector along
these three axes. In more familiar co-ordinate notation, we could just refer to the vector as
(ax, ay, az).
This vector has an amplitude given by applying Pythagorus’ theorem to the problem:
2
a  a x 2  a y 2  az 2
Finally, there are two ways of writing a vector – as a row vector, e.g.
a   ax
ay
az 
or as a column vector, e.g.
 ax 
a   a y 
 az 
In terms of the distance they represent, they are exactly the same thing. The only reason to
distinguish them is when you want to start multiplying them together.
3
If you’re using a multi-valued quantity that doesn’t behave like this in real life, then you can’t use a vector to
represent it.
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6.4 Multiplying Vectors
Vector multiplication is a bit strange – the most useful answer depends on what the vectors are
representing. There are three ways to multiply two vectors together, all of which give different
6.4.1
Vector Multiplication – The Dot Product
The first way to multiply vectors together is known as the dot product, scalar product or inner
product, and is written with a dot:
x  a . b  a b cos
Note that the dot product of two vectors is a scalar – that is, it has a magnitude, but no
direction, it’s just an ordinary scalar number. The dot product is defined as the product of the
magnitudes of the two vectors, times the cosine of the angle between them.
a

b
a .b
|b|
Figure 3 - The Vector Dot-Product
Note that the projection of a vector a along another vector b is then just a.b / |b|.
Expressing the vectors a and b in Cartesian co-ordinates shows that there is a very easy way to
calculate the dot product:





a . b   a x x  a y y  a z z  .  bx x  by y  bz z 








 a x bx x .x  a x by x .y  ax bz x .z  a y bx y.x  a y by y.y  a y bz y.z  az bx z .x  az by z .y  az bz z .z
 a x bx  a y by  a z bz
(Since in this case all the axes are orthogonal, and the cosine of 90 degrees is zero and the
cosine of 0 degrees is one, we have for example, x . y  x . z  y . z  0 and x . x  y . y  z . z  1 .)
In terms of row and column vectors, this is what you get when you multiply a row vector by a
column vector – but note, it has to be in that order, with the row vector first. Multiply them
together the other way around and you get something completely different4.
4
This one is sometimes called the ‘outer product’ and results in a matrix:
[continued on next page…]
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 ax

6.4.2
ay
 bx 
 
az  by   a . b  a x bx  a y by  az bz
b 
 z
Vector Multiplication – The Cross Product
The cross product of two vectors is the area of the parallelogram defined by the two vectors.
For example:
a
b
Figure 4 - The Vector Cross Product
The area of this parallelogram is given by a b sin  , where  is the angle between the two
vectors.
In vector language, areas have a direction (the direction perpendicular to the surface) so the
cross product of two vectors is another vector. To determine which direction, the “right-hand
rule” is used. Align the first vector with the thumb of your right-hand, the second vector with
your first (index) finger, place your second finger at right-angles to the thumb and first finger,
and the direction of your second finger gives the direction of the area. In the case shown
above, a  b is a vector going down into the paper away from you, b  a is a vector coming up
out of the paper towards you. (Cross-products are written a  b or sometimes as a  b .)
Note that a  b  b  a . The cross product is not commutative.
Again, in Cartesian co-ordinates, the cross product can be expressed as:

 
a  b  a x x  a y y  az z  bx x  by y  bz z




 x a y bz  az by  y  az bx  ax bz   z ax by  a y bx
6.4.3

Vector Multiplication – The Scalar Triple Product
This isn’t a new way to multiply vectors together, it’s just an interesting way to combine cross
and dot products when you want to multiply three vectors together. The scalar triple product
of three vectors a, b and c is written:
 ax 
 
 a y  bx
a 
 z
by
 a x bx

bz    a y bx

 a z bx
ax by
a y by
a z by
a x bz 

a y bz 

a z bz 
If you’ve not come across matrices before, don’t worry. See the chapter on matrices for more details.
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a.  b  c 
Since b  c is the area of the parallelogram with sides b and c, the scalar triple product is the
product of this area with the magnitude of vector a times the cosine of the angle between a and
the area. That’s just the volume of the parallelepiped which has the three vectors a, b and c as
the non-parallel sides. Clearly, therefore,
a.  b  c   b.  c  a   c.  a  b 
since they are all the same parallelepiped, but since b  c  c  b , we get:
a.  b  c   b.  c  a   c.  a  b   a.  c  b   b  a  c   c.  b  a 
6.4.4
Vector Division
You can add and subtract vectors easily, and you can multiply vectors in three different ways
(provided the two vectors have the same number of components), but you can’t divide them at
all. That might seem a bit odd – the cross-product of two vectors is another vectors, so if we
ab  c
then you might think that it would be possible, if given b and c, to work out a – and surely
that’s a form of division?
There are two problems with this idea. The first is that the cross-product of two vectors is
always perpendicular to the plane containing those vectors. So, if c is not perpendicular to b,
then there will be no value of a that satisfies this equation. That’s annoying, but it’s not in
itself a reason why division doesn’t work in all cases.
The more serious problem is that there is no unique answer. Given any value of b and c, any
vector a that is perpendicular to the vector c, and which has a length given by:
a
c
b sin 
where  is the angle between b and c will do. It’s not much use having an operation (division
in this case) that never gives a unique answer. It’s the same problem when trying to divide a
scalar by a vector using the dot product – there is no unique answer5.
6.5 An Application of Vectors: Adding Cosine Waves
All this has been rather theoretical so far – time for some examples of how vectors can actually
be useful. One common application of two-dimensional vectors is in representing cosine
waves. A cosine wave has the general formula:
5
The third, rarest form of vector multiplication does have a unique answer when the square matrix and one of the
component vectors is given, but there aren’t many practical uses for this.
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x(t )  A cos  t   
where A is the amplitude of the wave,  is the angular frequency and  is the phase of the
cosine wave at time t = 0. We often have to compute the sum of a large number of cosine
waves, all with different amplitudes and phases, but all at the same frequency. Vectors provide
a convenient way of doing this.
A cosine wave can conveniently be represented by a vector, with the amplitude of the vector
corresponding to the amplitude of the cosine wave, and the angle between the vector and some
arbitrary direction (usually taken to be the x-axis) representing the phase of the vector at time
t = 0.
To see how this works, consider the cosine wave A1 cos  t  1  . This can be written in
terms of the sum of a cosine wave and a sine wave both with a phase of zero at time t = 0,
using the trigonometric identity:
cos  A  B   cos( A) cos( B )  sin( A)sin( B)
Here, this gives:
x1 (t )  A1 cos 1  cos  t   A1 sin 1  sin  t 
If we represent this wave as a vector, with an x-component of A1 cos 1  , and a y-component
of  A1 sin 1  , this vector has a length of:
2
x1 (t ) 
 A1 cos(1 ) 2   A1 sin(1 ) 2
 A1 cos 2 (1 )  sin 2 (1 )  A1
and we can work out the angle between this vector and the x-axis using the dot-product:
cos   
 A1 cos(1 )
1 
 A1 sin(1 ) .  
 0   cos( )
1
A1
So, we have a way to represent this cosine wave as a two-element vector. The first component
(the x-component) of the vector is the amount of cos(t) in the cosine wave, the second (the
y-component) is the amount of sin(t).
To work out the sum of several cosine waves, all we have to do is express them in terms of the
amount of cos(t) and sin(t) they contain, add up the total amount of cos(t) and sin(t) in
the sum, and this will be the vector representation of the sum of all the cosine waves.
For example, the vector [2 0] represents the cosine wave x1(t) = 2cos(t), and the vector [1 1]
represents the cosine wave x2(t) =
2 cos  t   / 4  .
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(1,1)
(3,1)
(2,0)
Figure 5 - Adding Two Cosine Waves Using Vectors
Add up the components, and the final sum can be represented as:
x1 (t )  x2 (t )  3cos  t   sin  t 
which is a vector with an amplitude of
32  12  10 , and a phase of tan-1(1/3).
If it works for two cosine waves, it must work for any number of cosine waves as well. For
example, if we have a total of five different cosine waves, all with different amplitudes and
phases represented by the different vectors as shown below, the total received signal is
represented by the vector sum of these vectors:
Five
Cosine
Waves
Sum
Figure 6 - Adding Five Cosine Waves Using Vectors
This is a very common technique to work out the amplitude of the total received signal at a
radio antenna when there are a large number of rays arriving from the transmitter, all with
different amplitudes and phases.
6.6 Another Application of Vectors: Geometry
This isn’t as often used in communications engineering, but it does find some applications in
radio propagation modelling, and it’s often a neat way of working out some complex
trigonometric problems, so I think it’s worth introducing the subject. I’ll stick to three
dimensions for this section.
A three-component vector can readily represent a point in three-dimensional space. Therefore,
we should be able to express the equations of lines, planes, spheres and various other threedimensional objects in terms of vectors. For example, a point x lies on a spherical surface of
radius r centred around a point represented by a vector a from the origin provided:
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xa  r
A point x lies on a given line, provided:
x  a  kb
where a is any point on the line, b is a vector parallel to the line and k is a scalar.
A point x lies on a given plane, provided:
x.a  d 2
where a is the vector from the origin to the nearest point on the plane, and d is the shortest
distance from the origin to the plane.
This technique is particularly useful in determining the angle between two lines. For example,
suppose you have a line from the origin to the point (1,1,1), and another line from the origin to
the point (2,3,-1). What is the angle between these lines?
Since the dot product is the product of the lengths of the two vectors times the cosine of the
angle between them, and the dot product in this case is:
1
1
1 . 2
3
1  1 2  1 3  1 1   2  3  1  4
we can quickly work out that the angle between them is:

 a.b 
4
1
  cos 1 
  cos  2 2 2
2
2
2
a b
 1 1 1 2  3 1

 4 
  cos 1 


 42 

6.7 Problems
1) A vector representing a distance from the origin can be written as [1 2 2 4] in Cartesian coordinates. How far is this distance from the origin? What is the angle that this vector makes
with the four axes? What would be a unit vector in the same direction?
2) You’re given two vectors: a = [1 1 2] and b = [1 2 3]. What is their dot and cross products,
and what do these quantities represent?
3) Two radio waves arrive at a receive antenna at the same carrier frequency. One has twice
the amplitude of the other, and the phase difference between them is 45 degrees. Draw a
vector diagram illustrating this case, and hence determine the amplitude of the signal received
at the antenna.
4) Show that the scalar triple product of three vectors is zero if any two of the vectors are
parallel.
5) Prove that: a   b  c   b  a.c   c  a.b  . (This is a very useful result in electromagnetic
theory.)
2
2
2
2
6) Prove that for any two vectors a and b, a  b  a b  a.b .