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# CALCULUS II MATH 142

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This 7 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 142 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 16 views. For similar materials see /class/229548/math-142-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

MATH 142 A Brief Introduction to Vectors Spring 1995 While many quantities can be adequately described by a single number mass temperature barometric pressure speed time GPA population others seem to require two or even more numbers for a full description Some of these are velocity how fast and in which direction force how strong and in which direction multi vitamins how much of each of A B C D E etc TV audience share what is watching each channel animal or forest management the number in each age or height class income taxes the population in each tax bracket and so on lndeed models of a modern economy may involve thousands of numbers to give a complete description We will begin with the simple case of just two numbers but as we shall see many of the ideas carry over without dif culty to the more general situation You will quickly notice that this handout is not complete7many important ideas are explored only in the context of problems for you to work on Bring your questions about these to class In addition there will be the usual sorts of exercises on which you will get to practice computational skills A vector is just an ordered pair of real numbers v 11 Since boldface is not practical in handwritten work one often writes 7 or v or 2 instead There are a few special vectors that have symbols of their own the zero vector is 0 00 i 10 j 0 1 Of course it looks like there is a danger of confusing vectors with the coordinates of points in the plane but it turns out that the context always makes things clear and sometimes we even do want to have both meanings at the same timel Our rst example of a vector actually comes from geometry if P is the point 3y and Q is the point ry then the displacement vector 1 is the vector 33 7 c y 7 y All we are doing here is recording the net change in position of a particle if moves somehow not necessarily in a straight line from the point P to the point Q Note that since we are only considering net change in position we can forget about the points P and Q the same displacement could actually take place anywhere in the plane It is convenient to represent such a vector geometrically as an arrow from the point P which is called the tail to the point Q which is called the tip We can calculate the length of this arrow very easily by the Pythagorean theorem to be c 7 32 y 7 y2 More generally the magnitude or size of a vector v 11 is lvl V a2 b2 It is easy to see that the zero vector is the only one that has magnitude 0 It is conventional to put little hats on those vectors such as i and j that have magnitude 1 such vectors are called unit vectors By themselves vectors are not terribly interesting What makes them useful is that we can perform many of the usual arithmetic or algebraic operations with them Here are the two most basic ones Let us take v 11 and W c d 1 Vector addition v W a c b d 2 Scalar multiplication lf 7 is a real number rv rarb 3 Vector subtraction We abbreviate 71v as 7v Then W7v w7v In the following problems you will investigate these operations Basic properties Pretty much all the usual things work Show that v 0 v v 7v 0 and that vwwv Show that 0v 0 rvw rvrw and rsv rsv where s is another real number If u is yet another vector 1 show that u v W u v W We can also decompose vectors show that v 1 bj In fact in some texts usually in engineering it seems the notation a b is never used instead all vectors are written in their decomposed form ail bj The individual pieces 1 and bi are called the components of v Geometric interpretation Here we are working with displacement vectors Con I vince yourself that vector addition of two displacements v and W corresponds to the overall displacement from the tail of v to the tip of W when we place the tail of W at the tip of v What happens if we perform the displacements by placing the tail of v at the tip of W Suppose you put the tails of v and W at the same point Show that the arrow for v W also for W v runs from this point to the opposite corner of the parallelogram that has v on opposite sides and W on the other two sides This is called the parallelogram law for vector addition It turns out to be a useful way to describe how forces combine but for displacements the idea of one vector following after the other is better This is especially true for vector subtraction show how to represent W 7 v as an arrow by considering that this is the vector that you need to add to v in order to get W Describe the geometric meaning of rv what happens when 7 is negative Finally show what vector decomposition means geometrically Vectors representing forces lmagine two young Girl Scouts pulling a little red wagon loaded with Girl Scout cookies They are each pulling on the handle with a force of 10 units if you think about it we don t have near as good intuition about units of force as we do time and distanceel sure have no idea how much a newton is on a human scale and while ft lbs are of human scale they are bad terminology for understanding physics correctly How much force is exerted and in what direction if they are pulling at a 300 angle from the center line one on one side and one on the other How would your answer be different if the angle was 900 or 00 What if one is pulling at 100 and the other at 20quot Parametric equations and position vectors Up to now when we have thought about motion it has always been rectilinear that is the particle was constrained to move back and forth along a straight line In real life most motion occurs in two or even three dimensions In this case its motion is described by two separate functions of time 33t and yt The equations a 33t and y yt are called the parametric equations for the path of the particle and t is called the parameter Any such pair of equations in which a and y are expressed in terms of the same variable which need not be time are called parametric equations We can track a moving particle by its displacement from the origin at time t this is the position vector rt atyt ati ytj This is our first example of a vector valued function instead of putting in a number t and getting a value ft we are putting in a number and getting out a vector Fortunately as we shall see such functions are no harder to work with than ordinary functions Velocity vectors Imagine a particle traveling along a curved path in the plane Fix a time t and draw the vectors rt rt At and lAtrt At 7 rt for a moderate size At of course you ll want to use what you learned about vector subtraction and scalar multiplication above Do this for a smaller At and then for an even smaller one What seems to be happening geometrically as At approaches zero If you want to make things more speci c try using rt 15152 What curve is this particle following We shall actually de ne the derivative of the position vector or by another name the velocity vector to be vt rt limAtn01Atrt At 7 rt In the speci c example can you con rm that vt rt 1215 It is not to hard to see that in general for rt 305 2405 rti ytj we obtain rt ast 2405 ct yt Physically what this is saying is that velocity can be computed in each component separately In general the derivative of any vector valued function ft f1tf2t is simply computed by ft ftf t that is one computes the derivative in each component separately One of the most important facts about the derivative in the rst term was the microscope equation It holds for vector valued functions also where Af fa At 7 fa is the change in f and A is the change in t Dot product So far we have avoided multiplying vectors and there is a very good reason for this There just isn t any way to do it that behaves like ordinary multiplication There are however certain operations that behave a little bit like multiplication The rst of these is called the dot product or scalar product For v a b and W c d the dot product is v W ac bd The rst thing to notice is that the answer is no longer a vectorithe dot product of two vectors is a number that is a scalar Another peculiarity is that the dot product can be zero even when neither vector is zero consider for instance i j Nevertheless there are several properties that continue to be true for this funny product If 7 is a real number v and W are as above and u c f show that vWu vWvu v W W v and rv W v rW rv W Perhaps the most interesting result is that the dot product of a vector with itself is related closely to its length show that v v lvl2 Projection and orthogonal decomposition If you look back at the Girl Scout problem perhaps you can see that the pulling force that really counted was the part of it that was directed forward the part that pulled sideways was wasted so to speak What do we mean by talking about these parts The basic idea goes back to the decomposition of a vector that we mentioned earlier only instead of using i and 3 we will use perpendicular vectors that are more appropriate to our situation Suppose one of those Girl Scouts is sitting on a nice smooth slide if you must think like a physicist set a heavy block on a frictionless inclined plane If her mass is 20 Kg then the force of gravity F pulling her directly vertically has a magnitude of 196 newtons ie 196 Kg msec2 this comes from the equation F mg where g is the acceleration due to gravity directed vertically with a magnitude of 98 msec2 We can view this force as a vector sum of two separate forces one parallel to the slide which causes her to move and the other perpendicular to the slide pulling her right into the slide of course this is cancelled out by the resistance of the slide itself which has to do with the molecular structure of the metal and so on Let us call the parallel component FH and the perpendicular component FL We are simply saying that there is a decomposition F FH FL Such a decomposition is called orthogonal because the components are perpendicular orthogonal normal to one another It would be instructive at this point to make yourself some more diagrams that illustrate these vectors for slides at different inclinations 0 Our problem is to find a way to compute the magnitude of the effective force FH First show that the angle between F and FL is just 0 Then it is clear that lFil cosH why Finally how can you find lFHl Now that you know the force on the girl you can compute her acceleration velocity position and so on but since this isn t a physics class we won t do all that By the way this is how Galileo actually did his experiments on the effects of gravity he didn t just drop balls off towers The story doesn t end here Suppose we didn t know the angle 0 but we were in a situation where vectors F and D were known and we wanted to decompose F into components parallel and perpendicular to D This occurs for example if we want to calculate the work done by the force F in displacing an object by an amount D Since only the force in the direction of D contributes to the work W forcedistance lFHHDl cos 0lDl and we need to compute either FH or cos 0 You might wonder why force in a certain direction doesn t cause motion in exactly the same directionibut think about pushing a shopping cart whose wheels are locked in the wrong direction or the force of gravity on a person constrained to move along a slide According to the Law of Cosines the general property of vectors that lvl2 vv and the various other algebraic properties of the dot product mentioned above we have lFiDlZ lFl2lDl272lFHchos6 FiDF7DFFDD72lFHchos6 FFiFD7DFDDFFDD72lFHchos6 72FD72lFHchos0 which leads to the formula that shows why the dot product is so important FDlFHchos0 If we know F and D the left hand side is very easy to computeiit s just a little easy arithmetic The right hand side contains the geometric length and angle information and in particular from it we can compute the cosine of the angle between the two vectors As you might imagine one frequently can exploit this by making two hands arguments Then we see that the formula for work becomes very simple W lDl cos 0 FD But now we should get a little bit suspicious Dot products can turn out to be negative What would negative work mean The answer turns out to be fairly simple In all the pictures so far 0 has been an acute angle 0 S 0 S 7r2 and therefore cos 0 has not been negative But we might have a situation like this Then cos 0 lt 0 and the work comes out to be negative So our very first formula for work wasn t quite correct we should have said W ilFH l lDl where the sign is if FM is aligned with D and 7 if it is backwards Roughly speaking the idea is this if an apple falls to the ground gravity has done some work making the apple fall faster and faster or to be fancy giving the apple a nice dose of kinetic energy which you can feel if it bonks you But if we pick the apple up we are doing the work against gravity or again to be fancy we are giving the apple potential energy in this sense our moving the apple opposite to the force of gravity is undoing the work that gravity did If this all sounds ridiculous ask your physics professor for a better explanation Anyhow forgetting about work if F and D are just a couple of vectors we do get a nice formula for FM the vector projection of F on D If u is a unit vector in the direction of D then FH ilFH l cos 0 As we saw in the exercises f1 llDlD so lFlcosH lFHchosH F lt lDl D lDP D D Later on we shall find many applications for vector projection One should also note that the signed length of FM or cos 0 is sometimes called the scalar pro jection of F on D If someone asks you to compute the projection you should always check which of these quantities they mean Just one last remark once we have computed FH we can immediately compute FL how and hence obtain the orthogonal decomposition of F relative to D Whewl that was a long sectionidon t panic if it seems a little much at first Transformations So far we have seen functions that accept a number If as input and produce a vector ft as output We can go go one step further by considering functions that take vectors as input and produce vectors as output If we think of vectors as displacements from the origin and just look at what happens to their tips such a function amounts to putting in points of the plane and getting out points of the plane In other words we have a certain transformation of the plane This isn t really so strange For example one transformation that you are all familiar with is rotation say by 7r2 in the counterclockwise direction This transformation carries 0 to 0 i to j and j to 7i How can we describe such transformations One way is to give a formula The rotation transformation can be described by Tc y 7y 3 check that this worksl Can you produce a formula for re ection across the 3 axis How about a rotation by 7r6 counterclockwise Or one that pushes every point twice as far as it was from the origin but along the same line through the origin We will only deal with linear transformations for now these are ones that have formulas of the type Tc y a3 by 03 dy and include all the examples we have mentioned so far It turns out there is an ef cient way to represent these transformations We simply collect all the coef cients in a tidy little box called a matrix or an array if you happen to be a computer person In this case we have A Z b We also d 33 rewrite the vector v c y as the matrix v y the rst form is known as a row vector and the second as a column vectoriit is useful to switch back and forth as by 03 dy the result is a column vector but if we want to we can switch it back to row vector form a3 by 03 dy By the way Maple is really adamant about this it hates writing column vectors if it can possibly avoid it maybe because it takes up too much space on the screen It will insist that Av is a3 by 03 dy To remember how matrix multiplication works it is handy to think of it as really nothing more than a bunch of dot products Each row vector inside A gets dotted in its turn with the column vector v You should check that our first rotation example Then Tcy Tv is given by matrix multiplication Av Notice comes from using the matrix A l3 Tol Flipping across the c axis comes from 1 the matrix A 0 31 In the exercises you will get to work out some more examples So far the transformations that we have considered are so nice that we can understand how they work on the whole plane which by the way mathematicians 6 like to abbreviate by R2 all at once But some more complicated transformations are easier to understand if we look at how they transform just a piece of the plane such as the unit square With corners at 0 0 1 0 11 and 01

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