Vectors in two dimensions

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1 Vectors in two imensions Until now, we have been working in one imension only The main reason for this is to become familiar with the main physical ieas like Newton s secon law, without the aitional complication of vectors The time has come, however, to take this aitional step Many interesting physical effects are inherently more than oneimensional, an we woul like to learn about them Stuies have repeately shown that a poor grasp of vectors is one of the major causes of failure in introuctory physics courses The time an effort spent on eveloping a goo unerstaning of vectors now will be amply reware later on One of the unfortunate facts about this topic is that stuents come into first-year physics with wiely ifferent levels of backgroun in vectors nother fact is that university-level professors often on t want to spen much valuable class time reviewing vectors They prefer to get on to the real physics as soon as possible, an if you haven t alreay magically graspe enough knowlege about vectors, look out! This section, an its companion on vectors in three imensions, attempts to aress these problems You will fin that the emphasis is place on unerstaning the concepts involve, while the mathematics has been kept to a minimum Knowlege of trigonometry is assume, however You are really suppose to have learne this in high school If you are weak in this area, now is the time to review What is a vector? Suppose we are in city, an someone has tol us that a nearby city is some istance away Do we have enough information to fin? Of course not, because coul be at any point on a circle of raius centere at Here is a bir'seye view:?? In aition to the istance, we nee to know the

2 irection, in orer to fin One way of specifying the irection woul be to give the angle (the Greek letter theta) of, measure counter-clockwise from ue east Once both the istance an irection of the line from to are specifie, we can locate This is shown in the following figure: ue east Taken together, the istance an irection of the line from to is calle the isplacement from to, an is represente by the blue arrow in the above figure (The arrow-hea tells us that we are talking about the isplacement from to, an not to ) The isplacement is a classic example of a vector Definition: a vector is a quantity that has both magnitue an irection The magnitue of the isplacement vector from to is the istance from to It is important to remember that a vector is not completely specifie by its magnitue or irection alone; both are necessary nother familiar example of a vector is the velocity This vector points in the irection of motion, an its magnitue is the spee If only the spee is specifie, then the irection is unknown an the velocity is not completely specifie aseball pitchers often talk about having "goo velocity" Most of the time, what they are really talking about is just the magnitue of the velocity; their wor for the ability to etermine the irection of a pitch is "control" Often a pitcher will have "goo velocity" but "no control" He won't make it in either baseball or physics ecause of the irectional nature of a vector, it looks ifferent when viewe from ifferent irections In the resources for this lesson you will fin a movie illustrating the changing appearance of a vector when viewe from ifferent angles Components of a vector Returning to the isplacement vector from to, let's ask: "how much of the isplacement is in the easterly irection?" This is the same as asking how far is from, when viewe from ue south The answer is the length of the segment shae in green in the following figure:

3 ue east efine the angle between the above two vectors in this way: This length is referre to as the component of the isplacement vector in the easterly irection It is given in this example by cos There's nothing special about the fact that we foun the component of the isplacement vector in the easterly irection We can fin its component in any irection we like We simply specify the irection of interest by rawing a vector in that irection, an rop a line from the tip of the original vector perpenicular to this new vector For example, the component of the blue vector in the irection of the re vector in the following figure is cos α: α You might think that there is an ambiguity cause by the fact that we coul equally well 2π α However, cos ( 2 π α ) is the same as cos α, so there is no ambiguity For the value of α shown in the last two figures, cos α is positive This is not always the case, however For example, the re vector coul point like this: α The component in the irection of the re vector is still cos α, but this is now negative The absolute value of this number is the length of the green segment in the above figure

4 Specifying a vector in two imensions lthough a vector is efine as a quantity with magnitue an irection, it nee not be specifie irectly by the values of these two properties Returning to our iscussion of the isplacement vector, we coul equally well specify the isplacement by saying how far east an how far north is from That is, we coul specify the components of the isplacement vector in the easterly an northerly irections These are shown in green in the following figure: north a b east If the vector is specifie in this way, it is sai to be in component form This is completely equivalent with the polar form, in which the vector is specifie by its magnitue an its irection Of course, you can convert freely back an forth between the two forms This is a very common manipulation If you know the polar form (ie an are known), then the components a an b are given by a = cos an b = sin Conversely, if a an b are known, then an are given by = a 2 + b 2 an = arctan ( b / a ) You shoul not memorize these equations They will occur in many ifferent contexts, with ifferent variables an in ifferent notations However, you shoul be completely familiar with the ieas behin these equations, so that actually performing a conversion between polar an component form presents no problem The ieas, an not the mathematics, are the primary content of this lesson ition of vectors in two imensions Suppose we have a thir city, C, an suppose we know the istance an irection from to C (in aition to our previous knowlege of the isplacement vector from to ) Let's say we want to go irectly from to C What are the istance an irection? The first thing to notice is that if the three cities o not lie in a straight line, then the istance from to C will not be equal to the sum of the

5 istances from to an from to C lso, the irection will be relate in a complicate way to the two separate irections an istances: You can see, however, that the solution is easy if we work with the components of the isplacement vectors Let the components of the vector from to C in the easterly an northerly irections be be a an b, respectively Then it is obvious that the component of the isplacement vector from to C in the easterly irection is a + a, an in the northerly irection is b + b : north C Then the length of the isplacement vector from to C is ( a + a ) 2 + ( b + b ) 2 an its angle measure counter-clockwise from ue east is arctan ä å b + b ë ì ã a + a í This completely specifies the sum of the two separate isplacement vectors To get the sum of two vectors, you place them tip to tail an raw a thir vector from the tail to the tip of the whole thing In the following figure, the sum of the isplacement vectors from to an from to C is shown in re C b+b b b a a a+a 10 C east Note that the orer in which you o the sum is unimportant s the above figure shows, aing the vectors in the opposite orer requires you to slie them aroun parallel to themselves You get what looks like a fictitious path from to C,

6 going through some ghostly "fourth city" This is all right; what matters is the resulting path irectly from to C This is the same, no matter which orer the vectors are combine in Here is an important point that often causes confusion vector is specifie completely by its magnitue an irection The vector is the same, no matter where it is, as long as its magnitue an irection are the same The location of the vector is not part of its efinition You are free to "slie the vectors aroun" as long as you o not change their magnitue an irection Vector notation Instea of referring to a vector by a name like "the isplacement vector from to ", it is useful to have a symbol We enote a vector by an arrow over a letter like this: v P Different vectors will be istinguishe by ifferent letters The sum of two vectors is written u P + v P, for example Multiplication of a vector by a real number Suppose we a v P to itself We en up with a vector which is twice as long as the original, pointing in the same irection: P v It is natural to write v P + v P = 2 v P, where the righthan sie means a vector in the same irection as v P but twice as long Obviously, you can multiply a vector by any positive real number in the same way; for example, 15v P is a vector in the irection same irection as vp but 15 times as long Suppose we subtract vp from itself The result is obvious because when you subtract something from itself, you get zero: v P v P = 0 P (The righthan sie is the zero vector, a vector of length zero whose irection is unefine) This picture shows the operation of subtracting v P from itself: P v P v Subtracting v P is the same as aing a vector the same length as vp but in the opposite irection That is, P v v P v P = v P + ( v P ) Hence, if you multiply a vector by 1, you get a vector the same length as the original but in the

7 opposite irection Similarly, multiplying a vector by 15, say, gives a vector in the opposite irection an 15 times as long Multiplying a vector by the real number zero obviously gives the zero vector It s also useful to have a notation for the length or magnitue of a vector It is v P, an is a positive number or zero, by efinition Summary so far You now know all of the essential information about vectors: they have magnitue an irection; you can fin their component in any irection you choose; they can be ae together; they can be multiplie by a real number; there exists a zero vector lthough we have illustrate the above points using vectors in two imensions only, everything carries over into three imensions There is a section on vectors in three imensions in this course material You on't nee to go there now, if all you want to o is unerstan the concept of vectors, however The ot prouct The ot prouct (or scalar prouct, or inner prouct) of two vectors is efine to be the prouct of the lengths of the two vectors times the cosine of the angle between the vectors: a P b P = a P b P cos The ot prouct is just a number, in contrast to another kin of prouct calle the vector prouct or cross prouct, to be iscusse later Note that the length of a vector is just the square root of the ot prouct of the vector with itself: Unit vectors v P = v P v P Unit vectors are a hany way to specify irections Until now, we have specifie irections by saying things like ue east an ue north It is often useful to have a shorthan notation for these terms What we are now going to escribe is just notation - there is no more content to it than that

8 Let s make a vector which has length equal to one unit an points ue east We ll call this the unit vector in the x-irection an symbolize it by putting a hat over it: x ˆ Similarly, let s let the unit vector which points ue north be y ˆ (Other common notations for these unit vectors are î an ĵ or e ˆ x an e ˆ y ) Returning to our iagram which shows the components of a vector in the easterly an northerly irections, we fin that the vector can be expresse as the sum of multiples of the unit vectors: P v a x ˆ b y ˆ y ˆ x ˆ where a an b are relate to the magnitue an irection of vp as before Vectors are particularly easy to manipulate when written like this For example, if we have another vector then w P = cx ˆ + y ˆ, v P 2 w P = ( a 2 c ) x ˆ + ( b 2 ) y ˆ Suppose we know the components of two vectors Can we easily calculate their ot prouct? The answer is yes The ot prouct of the above two vectors turns out to be just the sum of the proucts of their components: v P w P = ac + b To check this, consier the case where w P points in the x ˆ irection (If it oesn t, then convert everything to a new set of unit vectors in which it oes) This means we may set =0 Remember that a x ˆ is a vector in the x ˆ -irection which has length a In symbols, v P = ax ˆ + by ˆ, P v a c b w P

9 eginning with the efinition of the ot prouct, we fin as claime v P w P = v P w P cos = á v P cos é c = ac, Polar coorinates an unit vectors This section is inclue here mainly for future reference You can safely skip it when you are reaing about vectors for the first time The unit vectors we have just iscusse are most appropriate when we are using rectangular coorinates That is, we are specifying the location of any point in the plane by stating its x- an y-coorinates: y x y ˆ x ˆ However, it is useful in many applications to specify points by their polar coorinates These are r, the istance from the origin to the point, an, the angle measure clockwise from the x- axis: r ˆ In this case, the appropriate unit vectors to use are r ˆ an ˆ, as shown in the above figure The former points in the irection of increasing raial coorinate r, while the latter points in the irection of increasing angle Note that these unit vectors are not fixe Their irection epens on where they are That is, they remain at right angles to one another, but both point in ifferent irections epening on the value of Compare the next figure with the previous one: r ˆ In contrast, the unit vectors x ˆ an y ˆ are fixe, r ˆ r ˆ y ˆ x ˆ

10 once an for all For this reason, it is often useful to express r ˆ an ˆ in terms of x ˆ an y ˆ It woul be an excellent exercise for you to show that the relations are r ˆ = x ˆ cos + y ˆ sin ˆ = x ˆ sin + y ˆ cos Sometimes it s also useful to be able to go the other way The inverse relations are x ˆ = r ˆ cos ˆ sin y ˆ = r ˆ sin + ˆ cos The velocity an acceleration vectors Suppose we have a boy which moves from place to place Then its isplacement vector will be a function of time: P r ( t ) = x ˆ x ( t ) + y ˆ y ( t ) Don t let the notation confuse you; the things with the hats are just the usual fixe unit vectors, an x(t) an y(t) are the components of the isplacement vector They are functions of time The velocity vector is just the time erivative of the isplacement vector: v P ( t ) = P r ( t ) The unit vectors we are using here on t epen on time, so the velocity vector in component form is v P ( t ) = x ˆ x( t ) + y ˆ y( t ) Similarly, the acceleration vector is a P ( t ) v P ( t ) = x ˆ 2 x ( t ) 2 + y ˆ 2 y ( t ) 2 Velocity an acceleration in polar coorinates The expressions in the last section are given in rectangular coorinates Sometimes, particularly when we are ealing with circular motion, it is useful to have expressions for position, velocity an acceleration in polar coorinates The isplacement vector is P r ( t ) = r ˆ r ( t ) Of course, this has no component in the - irection In orer to fin the velocity, we have to ifferentiate an take into account the fact that r ˆ changes irection as the position of the particle changes:

11 r P ( t ) = r ˆ r + r ˆ r Let s eal with this aspect first Going back to our expression for r ˆ in terms of the fixe unit vectors x ˆ an y ˆ, an ifferentiating, gives r ˆ = x ˆ ω sin + y ˆ ω cos We have use the common short-han notation for the angular velocity ω = Notice that the right-han sie is just proportional to ˆ : r ˆ = ω ˆ This says that r ˆ just twists aroun in the ˆ - irection Inserting into our earlier expression gives the final result v P ( t ) = r N r ˆ + r ω ˆ Here, we have use the shorthan notation N = (Note: if you can t see the prime clearly, use Reaer s magnification tool to increase the magification) The next figure shows a picture of the above equation It gives the ecomposition of the velocity into a raial part (the first term) an an angular part (the secon term) rω ˆ P r ( t ) P v ( t ) r N r ˆ path We ifferentiate again to fin the acceleration: v P ( t ) = r O r ˆ + r N r ˆ + ( r N ω + r ω N ) ˆ + r ω ˆ In the same way as we worke out the time erivative of r ˆ before, we can show that ˆ = ω r ˆ

12 little re-arranging then shows that the acceleration is a P ( t ) = r ˆ á r O r ω 2 é + ˆ 1 r á r 2 ω é For an application of these formulas, see the section on circular motion Vector fiels fiel is anything which is efine at all points in some region of space For example, the temperature of the air in a room has a value at each point in the room The values coul be ifferent, or they coul be the same In the case of temperature, the value of the fiel at any given point is a single number Such a fiel is calle a scalar fiel Mathematically, a scalar fiel is specifie by a single function of the coorinates, written F ( P r ) Notice that P r is a vector (specifying the point in the room, for example), while the value F ( P r ) of the fiel at that point is a single number There are other kins of fiels besies scalar fiels The next most complicate kin is a vector fiel s you might expect, that s a fiel whose value at each point is a vector Such a thing is specifie mathematically by as many functions as there are spatial imensions In two imensions, it s written F P ( P r ) = x ˆ F x ( P r ) + y ˆ F y ( P r ) The x-component of the vector fiel is specifie by a single function F x ( P r ), while the y-component is specifie by another function F y ( P r ) force fiel is one type of vector fiel Suppose the force on a boy epens on where the boy is locate Then the set of the force vectors at all points in space is a fiel Examples inclue the gravitational fiel an the electric an magnetic fiels, all of which we will stuy later There are other, higher kins of fiels For example, a fiel whose value at each point is a matrix (an array of numbers) is calle a tensor fiel These fiels are stuie in later courses Time-epenent fiels Fiels often epen on time time-epenent scalar fiel is written F ( P r, t ) Here is a movie showing a time-epenent scalar

13 fiel The value of the fiel at a point in the horizontal plane is given by the vertical coorinate of the surface above that point time-epenent vector fiel is written F P ( P r, t ) Here is a movie an another movie showing time-epenent vector fiels

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors

Math Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+