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 time worrying about. © 2006 University of York Page 1 23/02/2010 Communications Engineering MSc - Preliminary Reading 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.) 6.2 Adding and Subtracting Vectors 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. © 2006 University of York Page 2 23/02/2010 Communications Engineering MSc - Preliminary Reading 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. © 2006 University of York Page 3 23/02/2010 Communications Engineering MSc - Preliminary Reading 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 answers. 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…] © 2006 University of York Page 4 23/02/2010 Communications Engineering MSc - Preliminary Reading 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. © 2006 University of York Page 5 23/02/2010 Communications Engineering MSc - Preliminary Reading 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 had three vectors such that: ab 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. © 2006 University of York Page 6 23/02/2010 Communications Engineering MSc - Preliminary Reading 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) = © 2006 University of York 2 cos t / 4 . Page 7 23/02/2010 Communications Engineering MSc - Preliminary Reading (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: © 2006 University of York Page 8 23/02/2010 Communications Engineering MSc - Preliminary Reading xa 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 . © 2006 University of York Page 9 23/02/2010

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