Vectors 1. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Transcription

1 Vectors 1 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

2 Launch Mathematica. Type <<Mathetic`vecpack` Instructions for Getting Started hold down the shift key, and press the return key. Wait for Mathematica s response. (Note: be sure to use the ` symbol rather than the '. You may need to hunt for it on your keyboard: on most, it s in the top left corner.) This essential first step sets up Mathematica for this module. If you omit this bit, the special commands (see below) will not work. Mathematica Commands The following Mathematica commands may be useful to you in this module. Commands that come with Mathematica: N, Plot, ArcTan Special commands for this module: ShowVector, VectorSum, GiveQuestion, LastAnswer, ScalarMultiply, ToComp, ToMagDir, Magnitude For information on, say, the ShowVector command, type?showvector hold down shift, and press return. And, if you wanted a list of all the commands containing the word Vector you could type?*vector*

3 Introduction to vectors Vector and scalar quantities Some physical quantities, such as mass and time, can be completely represented by simple numbers, for example: 30 kilograms ; 7.5 seconds. For other types of quantities, such as force and acceleration, it s important to specify two things: i. how large the quantity is, and ii. in what direction it is acting. We might, for example, talk about: A force of 50 Newtons acting on a bearing of 120 degrees. Quantities which have direction as well as magnitude are called vector quantities. In contrast, those which have magnitude only (like mass and time) are called scalar quantities. Other physically important vector quantities include: Displacement (the answer to the question: how far away, and in what direction? ) Velocity (the answer to the question: how fast, and in what direction? ) Acceleration (the answer to the question: how is the velocity changing? ) Note that the magnitude of a displacement is distance, and the magnitude of a velocity is speed. Directed line segments The most natural way to represent a vector quantity diagrammatically is as a line segment with an arrow on it: The length of the line segment represents the vector s magnitude, in whatever units we're using: Newtons, metres per second, etc. The direction of the arrow represents the vector's direction. The vector shown above is 5 units long and it acts at an angle of 40 degrees anticlockwise from the direction East.

4 QUESTION: In maths, it s usual to measure angles anticlockwise from a horizontal, right-pointing baseline. How did this convention arise, do you think? Give this problem a little thought before checking the answer at the bottom of the page 1. Experiment 1: Drawing vectors with Mathematica Preparatory reading This experiment introduces some Mathematica commands for drawing vectors of given magnitude and direction. The vector in the diagram above has a magnitude of 5 units and a direction of 40 degrees. We are going to call this the vector {5, 40}, using braces (curly brackets, that is). This is not standard mathematical notation, but we need a neat way of representing vectors if we want Mathematica to process them. Braces, left { and right }, are an important part of Mathematica s language; they are used whenever we want to collect things together into a single bigger thing. Mathematically, that thing is a type of set, but in Mathematica it s called a list. There are a number of notations for vector quantities. In this module we will refer to any general vector using a lower case letter in bold type: v is our favourite name ( v for vector ). When writing by hand, you indicate a vector by underlining, either with a straight line, or some people use a wavy line like the tilde symbol ~. Another notation is to put an arrow over the top of a letter to indicate that it s a vector, v r for example; we will reserve this notation for displacement vectors (see Experiment 2). 1) Type in the following Mathematica input, hold down the shift key, and press the return key (which may be called enter on some keyboards): ShowVector[{5, 40}] This should produce a diagram exactly like the one shown above, that is, a vector of magnitude 5 units and direction 40 degrees anticlockwise from East. 2) Try the following: ShowVector[{15, 40}] ShowVector[{5, 110}] ShowVector[{5, 110}, ShowEast -> False] That last statement is an example of specifying an option to a Mathematica command ( set the value of ShowEast to False ). If you need to familiarise yourself with this vector notation, try plotting vectors in the compass directions N, W, S, E, NE, etc. Investigate the behaviour of ShowVector when the option AngleUnit->Rad is applied. 1 ANSWER: Turning anticlockwise from a right-pointing horizontal line means turning from the x-axis to the y- axis. Since x comes before y in the alphabet, this seems logical: at least to a mathematician! It also means that sine and cosine functions start off positive, which is tidy. But basically this is just a convention. Angles going clockwise from the right-pointing horizontal line are counted as negative: for example, a direction of 90 degrees means due South. Map-makers and navigators use a rather different convention: they measure angles ( bearings ) clockwise from North.

5 3) As you ll discover much more later on, vectors are added by connecting them nose-to-tail. This command adds together the vectors {1, 90}, {1, 45} and {2, -90}: VectorSum[{{1,90}, {1,45}, {2,-90}}] Note the extra pair of braces collecting the three vectors together; the vectors are shown labelled A, B, C etc. in the order they were given as input. 4) Use this command to: (i) add together four vectors to form a square; (ii) add together three vectors to form an equilateral triangle; (iii) add together three vectors to form an isosceles triangle. Do this for other shapes rectangle, trapezium, pentagon, six-pointed star, and so forth. 5) Consider: (i) Does the order in which the vectors are specified change the resulting shape? (ii) What doesn t change if you change the order? What does this tell you about the nature of vector addition? Make a note of your findings; we ll come back to them in a later experiment. This section uses this module s special functions. If they fail to work, try going back to the Instructions for Getting Started at the beginning, Experiment 2: Negative vectors and equal vectors Preparatory reading Displacement vectors Consider two points, A and B. The displacement vector from A to B is called, not surprisingly, the vector AB. In textbooks, vector AB will be written in bold type like this, or with a direction arrow over the top as in the diagram below. When you write it down, you should also use an arrow. The computer won t easily let us use either convention on the screen but it should be clear from the context when we mean a vector.

6 Remember that the vector AB is defined only by its magnitude and direction: if we move it away from A and B, it s still the same vector, and we can still call it AB: Negative vectors Given a displacement vector AB connecting the points A and B, how does it relate to the vector BA? The magnitudes must be equal, but the directions are opposite. In fact, we can say that BA is the negative vector of AB, that is, BA = AB. Equal vectors Two vectors are equal if they have the same magnitude and the same direction. They do not have to match in any other way and, in particular, they don t have to be acting in the same place. In the following diagram, the displacement vector from Farnby to Lowford is equal to the displacement vector from Highton to Bixleigh, even though they re in different places:

7 If two vectors have the same magnitude, but not the same direction, they are not equal. The two vectors in the diagram below are unequal, although they have equal magnitude, because they have different directions. We can instruct the VectorSum command to plot diagrams of displacement vectors by applying the option LabelStyle->AB as follows: VectorSum[{{1.7,55}, {1.3,29}, LabelStyle->AB] 1) In the following command what values for m and d will define the negative vector of {2.3, 47}? (That is, the vector such that the vectors marked AB and CB in Mathematica s diagram appear coincident): VectorSum[{{2.3,47}, {m,d}}, LabelStyle->AB] 2) The following command will generate the figure of a six-pointed star: VectorSum[{{1,0}, {1,-60}, {1,60}, {1,0}, {1,120}, {1,60},{1,180}, {1,120}, {1,240}, {1,180}, {1,300}, {1,240}}] How many different vectors would be needed to construct an n-pointed star, where n = 5, 6, 7,...? This section uses this module s special functions. If they fail to work try going back to the Instructions for Getting Started at the beginning,

8 Practice Questions This module includes a feature which allows you to get Mathematica to generate practice questions and their answers. There are two sets of questions on equality of vectors. To generate a question from the first set, type GiveQuestion["equal vectors"] not forgetting to shift-return. To generate the answer for checking, type LastAnswer["equal vectors"] You can do this as often as you want: the questions are randomly generated, and repetitions should be rare. Note: You don t need to retype these commands for another question; simply click the mouse on the command and shift-return. The other set of questions is accessed using GiveQuestion["equal magnitudes"] Experiment 3: Multiplication by a scalar Preparatory reading If we have some vector v of magnitude m and direction d degrees, that is v = {m, d} in the notation we ve been using, what do we mean by multiplying v by a scalar, say 3 times v = 3 times {m, d}? If the scalar is greater than zero then what we do is to leave the direction unchanged and multiply the magnitude by 3, so 3 v = {3m, d}. Although we ve written 3 v just like ordinary multiplication, it s important to remember that even though the notation is the same the rule is quite different (for example, try typing 3*{m,d} in Mathematica what happens?) Furthermore, what does it mean to multiply a vector by a scalar less than zero? Or equal to zero? 1) What happens if we multiply a vector by a scalar which is negative? For example we multiply the vector {4, 132} by -3.5? If this worked like the case for positive scalars then we d leave the direction unchanged and multiply 4 by What does ShowVector make of a vector with negative magnitude? Can you see what s going on? Use the special function ScalarMultiply to check your hypothesis: ScalarMultiply[3, {m,d}] ScalarMultiply[-3.5, {4,132}]

9 2) What happens if we multiply a vector by the scalar zero? According to the rule, the magnitude would be zero, but what then would be the direction? What does ScalarMultiply do? What does ShowVector do if asked to draw a vector of magnitude zero? 3) It may be useful to compare scalar multiplication with arithmetical multiplication. The latter can be defined as repeated addition, for example 4 3 = , and scalar multiplication of vectors can be considered as repeated vector addition. For example, ScalarMultiply[- 3.5, {4,132}] is equivalent to: VectorSum[{{-4,132}, {-4,132}, {-4,132}, {-2,132}}, ResultQ->True] This is getting a little bit ahead of the game, though, since vector addition is the subject of the next experiment. Post-experiment reading Here are all the possible cases for scalar multiplication, s times v: if the scalar s is greater than zero, the outcome is a vector of the same direction with magnitude s times the original. If s is less than zero, the outcome is a vector of magnitude s times the original with a reversed direction (the original direction plus, or minus, 180 degrees). And if s = 0, the outcome is the zero vector. The zero vector, written as 0 (a bold nought, or underlined if you re writing it by hand) is the vector with zero magnitude and indeterminate direction (that is, it just hasn t got a definite direction). Practice Questions There is one set of questions on scalar multiplication. To generate a question type GiveQuestion["scalar multiply"] not forgetting to shift-return. To generate the answer for checking, type LastAnswer["scalar multiply"] You can do this as often as you want: the questions are randomly generated, and repetitions should be rare. Note: You don t need to retype these commands for another question; simply click the mouse on the command and shift-return. Experiment 4: Vector addition Preparatory reading Consider the following problem: B is 5 metres due East of A. C is 12 metres due South of B. How far away from A is C, and in what direction? We could recast this in vector terms as follows:

10 Vector AB has magnitude 5 metres and direction 0 degrees. Vector BC has magnitude 12 metres and direction 90 degrees. find the magnitude and direction of vector AC. Or, more simply: What single displacement vector does the job of {5, 0} followed by {12, 90}? You ll notice that, according to this definition: AC = AB + BC. The statement of vector addition called the Triangle Rule can be summed up like this: (i) (ii) (iii) link the vectors head to tail; join the tail of the first to the head of the second: this is the answer, and then use trigonometry to calculate its magnitude and direction. 1) Draw a diagram to represent the addition of the vectors AB = {5, 0} and BC = {12, 90} by the Triangle Rule. Use your knowledge of triangles to calculate the magnitude and direction of the vector AC. 2) You can use the VectorSum command to check your answer: VectorSum[{{5,0}, {12,-90}}, LabelStyle->AB, ResultQ->True] The sum of vectors is called the resultant vector (hence we called this extra option ResultQ). Notice that the resultant is indicated by a double-headed arrow; this is just a useful convention, it does not mean that the resultant is any special kind of vector. 3) There s another way to look at vector addition geometrically, called the Parallelogram Rule. This command shows the diagram for the vectors of part 1: VectorSum[{{5,0}, {12,-90}}, AddRule->Parallelogram] Can you see how the parallelogram rule works? Try VectorSum with various inputs. The answer is given in the Post-experiment Reading. 4) The fact that two vectors can be added together means that we can add any number of vectors together. And we ve been doing this often in previous experiments with VectorSum. For example, here s the first use of VectorSum in Experiment 1, part 3, now with the ResultQ option applied: VectorSum[{{1,90}, {1,45}, {2,-90}}, LabelStyle->AB, ResultQ->True] Try out some further VectorSum commands of your own. Can you describe what s going on mathematically? An explanation is given in the Post-experiment Reading.

11 5) What about vector subtraction? It s very easy to convert any subtraction v - w into an equivalent addition can you see how? Post-experiment reading The procedure for the Parallelogram Rule introduced in the experiment is: (i) link the two vectors tail to tail ; (ii) complete the parallelogram formed by the vectors (the dashed lines in Mathematica s diagram); (iii) the sum of the two vectors is the diagonal of the parallelogram. The Triangle and Parallelogram Rules are exactly equivalent. There s a big problem with doing vector addition in magnitude direction form, which is to do with finding lengths of sides and angles in triangles. The first example we chose was easy to solve because the triangle was right-angled; in general this will not be the case and the trigonometrical calculations required will be rather harder. There is in fact an alternative way to represent vectors, called component form, which makes vector addition very easy to do; more on this in the next experiment. The fact that two vectors can be added together means that we can add any number of vectors together, by a process of repeatedly adding up all the connected pairs of vectors until we get to the single, resultant vector. For example, consider the following diagram: Here, we have three position vectors that we wish to sum. Suppose we decide first to add BC and CD, then: AB + BC CD 3 = 1 AB BD 3 = AD add add Write down similar sets of equations for four, five, six and more position vectors. Convince yourself that the order in which vectors are paired doesn t change the answer; this is the associative property of vector addition which we ll come back to in Experiment 6. Vector subtraction is the same process as addition, since for any subtraction v w we may write: v w = v + ( w)

12 and we know that the vector w is one having the same magnitude as w but reversed direction (rotation by +180, or 180, degrees; see following figure). Experiment 5: Components Preparatory reading Adding two vectors produces the single vector that does the job of the original two. We re now going to reverse the process and ask: what two vectors are equivalent to a single vector? Specifically: given a vector v, what horizontal vector can be added to what vertical vector to make v? For example, in the diagram below, the single vector on the left is equivalent to the two perpendicular vectors on the right (according to the Parallelogram Rule): These two vectors are called the horizontal and vertical components of v. Together they re the perpendicular components of v, and we can describe every vector uniquely in terms of its components. We will tend from now on to express vectors in component form instead of magnitude direction form. The process of finding the components of a vector is sometimes called resolving the vector into its components. 1) In the following diagram, use trigonometry to find the magnitudes of v x and v y, the horizontal and vertical components of the velocity vector v = {7, 60} m/s.

13 2) You can check your answer using the command ToComp[{7,60}] Try using ToComp with input vectors having directions 0, ±90, ±180, ±45, ±135, etc. degrees. 3) The commands ShowVector and VectorSum each permit the option setting VectorInput->Components. This causes each vector to be read as an x- and a y- component instead of a magnitude and a direction. Try the following: ShowVector[{3,4}, VectorInput->Components] VectorSum[{{3,4}, {4,-7}}, ResultQ->True, LabelStyle->AB, VectorInput->Components] Explore further how vector addition works in component form, using the VectorSum command. Write a brief account of your findings. 4) So far, we have not displayed vectors in any particular coordinate system. If you want to display vectors in an x-y coordinate grid there is an Axes option for all the plotting commands we ve been using. For example: ShowVector[{5,53}, Axes->True] VectorSum[{{5,53}, {4,-90}}, ResultQ->True, LabelStyle->AB, Axes->True] In the Post-experiment Reading there is a discussion of the relationship between vector representations and coordinate systems. 5) Investigate scalar multiplication in component form. How does it work when the scalar is greater, less than or equal to zero? If you want, use the ScalarMultiply command with its input format set to Components: ScalarMultiply[2, {3,5}, VectorInput->Components] Post-experiment reading Vectors in component form For a vector v = {r, θ}, the perpendicular components are: v x = r cos θ, v y = r sin θ.

14 There is a standard notation for vectors in component form: in fact, there s more than one. The first one we ll mention looks like this: It is called column notation. In Mathematica we will generally use a curly bracket notation for vectors in component form, {3.5, 6.06} for example, as we also did for magnitude-direction form. This is potentially confusing: it is important to keep in mind that curly brackets are a general way within Mathematica to group things together into lists, and the meaning of a list depends entirely on the context in which it is used. Position vectors In circumstances where we ve defined a pair of axes and an origin, O, the vector OA (A s displacement from the origin) is called A s position vector. It tells us where the point A is. For a position vector, the coordinates of A, (x, y) say, are equivalent to the components of the vector x OA,. And the polar coordinates of A, r = x 2 + y 2 and θ = arctan(y/x), are equivalent to the y magnitude and direction of OA. Vector addition in component form When the vectors x 1 and y 1 x 2 are added, the resultant is the vector y 2 x 1 + x 2. For example, y 1 + y = Scalar multiplication in component form With vectors in component form we can perform scalar multiplication simply by multiplying both components individually by the scalar, no matter whether the scalar is positive, negative or zero. There s an interesting point to do with the zero case: we said earlier that the zero vector, 0, has zero magnitude and indeterminate direction how is it that we can say definitely that 0 is a vector whose components are all zero?

15 Practice Questions There is a set of questions for you to practice converting vectors from magnitude-direction to component form. Use GiveQuestion["components"] to generate a question, and to check your answer: LastAnswer["components"] Experiment 6: Addition by components Preparatory reading There s an alternative notation for vectors in component form which can be very useful. The idea is this: let i be a horizontal vector of length one unit, and let j be a vertical vector of length one unit. i and j are both unit vectors: their magnitude is one unit. Then, for example, the vector is the vector sum the resultant of three lots of i and two lots of j, as shown in the diagram: 3 2 And we can write it as: 3i + 2j The unit vectors i and j are called the Cartesian basis in two dimensions. 1) The Cartesian basis notation for vectors can be used in Mathematica by defining the two unit vectors: i={1, 0}; j={0, 1}; Then, we may express any vector as a sum of multiples of i and j. For example: ShowVector[3i + 2j, VectorInput->Components, Axes->True] VectorSum[{3i + 2j, -2i, -i - 2j}, VectorInput->Components]

16 VectorSum[{i, i, i, j, j}, VectorInput->Components, ResultQ->True] 2) These example commands show some particular vector additions in component form; you ve probably got a good idea what the general rule is. Use VectorSum to test your hypothesis with different combinations of vectors the answer is given in the Post-experiment Reading. 3) Describe the vectors given by ai + 3aj where a is any constant. What s the relationship between the vectors ai + bj and bi + aj where a and b are any constants? 4) Use VectorSum to examine the properties of vector addition: (i) is vector addition commutative? That is can we say that for any two vectors, v and w, v + w is the same as w + v? (ii) Is vector addition associative? That is, does (a + b) + c = a + (b + c) for any vectors a, b and c? Do these findings confirm some of the ideas you had during Experiment 1, part 5? Post-experiment reading The rule for addition of vectors in component form is simply to add together the components themselves. Thus: or equivalently: Vector addition is both commutative, for any vectors v and w, and associative, for any vectors a, b and c. a + c = a + c b d b + d (ai + bj) + (ci + dj) = (a + c)i + (b + d)j. v + w = w + v (a + b) + c = a + (b + c) Experiment 7: Converting back, and the modulus of a vector Preparatory reading The magnitude of a vector is such an important thing that there are several different notations for it. For a vector v, the magnitude can be written as v, like a normal variable (or without underlining, if you re writing by hand), or as v, using the modulus sign. In fact, the name modulus is often used instead of magnitude. The modulus (absolute value) of a number and the modulus of a vector are

17 related quantities can you see why? The modulus notation is useful because it can be applied over complicated expressions, for example: a + 3v 2 3 w + b 5 means the modulus of the result of adding together all the vectors inside. Note that, although the modulus signs look like brackets, they definitely don t behave like brackets, for example: in general. 3 a + b 3a + 3b 3 a + 3 b The magnitude, or modulus, of the vector x i + y j, for any x and y, is x 2 + y 2. 1) Find the magnitude and direction of the vector 3i + 4j (you ll need to use Pythagoras Theorem and some trig). Check your answer using the command: ToMagDir[{-3, 4}] Recall the companion command for converting the other way: ToComp[{13, }] There is also a special command for calculating the magnitude only: Magnitude[{-3, 4}] 2) The main difficulty in converting to magnitude direction form is handling the arctan function correctly to arrive in the right quadrant. Mathematica s ArcTan command uses the conventional principal branch for arctan, as you can see by doing: ArcTan[-Infinity] ArcTan[Infinity] Plot[ArcTan[x], {x, -100, 100}] (Yes, Mathematica is cool about infinities). The output is in radians; you can always multiply by the factor 180/Pi if you prefer to think in degrees, and you may then need to use the N command to get things into decimals. In fact, ArcTan is cleverer than that, because it also knows about solving the quadrant problem: if ArcTan is given two arguments, ArcTan[x, y] then it performs arctan(y/x) applying the right sign for the quadrant in which the point (x, y) lies. The ToMagDir command uses this feature in its workings. 3) What can you say about the directions of vectors which lie along the x-axis or the y-axis? What about those having equal components (positive or negative), or components equal in magnitude but opposite in sign?

18 Practice Questions There is a set of questions for you to practice converting vectors from component form to magnitude direction form. Use GiveQuestion["to magdir"] to generate a question and LastAnswer["to magdir"] to check your answer.