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Vector Algebra - University of Utah

CHAPTER 13 Vector Basic ConceptsAvectorVin the plane or in space is an arrow: it is determined by its length, denotedjVjand itsdirection. Two arrows represent the same Vector if they have the same length and are parallel (see ). We use vectors to represent entities which are described by magnitude and direction. For example,a force applied at a point is a Vector : it is completely determined by the magnitude of the force and thedirection in which it is applied. An object moving in space has, at any given time, a direction of motion,and a speed. This is represented by the velocity Vector of the motion. More precisely, the velocity vectorat a point is an arrow of length the speed (ds=dt), which lies on the tangent line to the trajectory. Thesuccess and importance of Vector Algebra derives from the interplay between geometric interpretationand algebraic calculation.

coordinate system has been chosen: a point O, the origin, and two perpendicular lines through the origin, the x- andy-axes. A vectorV is determined by its length, j V and its direction, which we can describe by the angle θthat V makes with the horizontal (see figure 13.4). In this figure, we have realized V as the vector OP ~ from the origin ...

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Transcription of Vector Algebra - University of Utah

1 CHAPTER 13 Vector Basic ConceptsAvectorVin the plane or in space is an arrow: it is determined by its length, denotedjVjand itsdirection. Two arrows represent the same Vector if they have the same length and are parallel (see ). We use vectors to represent entities which are described by magnitude and direction. For example,a force applied at a point is a Vector : it is completely determined by the magnitude of the force and thedirection in which it is applied. An object moving in space has, at any given time, a direction of motion,and a speed. This is represented by the velocity Vector of the motion. More precisely, the velocity vectorat a point is an arrow of length the speed (ds=dt), which lies on the tangent line to the trajectory. Thesuccess and importance of Vector Algebra derives from the interplay between geometric interpretationand algebraic calculation.

2 In these notes, we will define the relevant concepts geometrically, and let thislead us to the algebraic +Figure +WNewton did not write in terms of vectors, but through his diagrams we see that he clearly thought offorces in these terms. For example, he postulated that two forces acting simultaneously can be treatedas acting sequentially. So suppose two forces, represented by vectorsVandW, act on an object at aparticular point. What the object feels is theresultantof these two forces, which can be calculated byplacing the vectors end to end (as in figure ). Then the resultant is the Vector from the initial pointof the first Vector to the end point of the second. Clearly, this is the same if we reverse the order of thevectors. We call this thesumof the two vectors, denotedV+W.

3 For example, if an object is movingin a fluid in space with a velocityV, while the fluid is moving with velocityW, then the object moves(relative to a fixed point) with velocityV+ Concepts187 Definition ) Avectorrepresents the length and direction of a line segment. Thelengthis denotedjVj. Aunitvector Uis a Vector of length 1. Thedirectionof a vctorVis the unit vectorUparallel toV:U=V= ) Given two points P;Q, the Vector from P to Q is denoted~ ) Addition. Thesum, orresultant,V+Wof two vectorsVandWis the diagonal of the parallelogramwith sidesV, ) Scalar Multiplication. To distinguish them from vectors, real numbers are calledscalars. If c is apositve real number, cVis the Vector with the same direction asVand of length cjVj. If c negative, it isthe same, but directed in the opposite note that the vectorsV,cVare parallel, and conversely, if two vectors are parallel (that is, theyhave the same direction), then one is a scalar multiple of the ;;Q;Rbe three points in the plane not lying on a line.

4 Then( )~PQ+~QR+~RP=0:From figure , we see that the Vector ~RPis the same line segment as~PQ+~QR, but points in theopposite direction. Thus~RP= (~PQ+~QR).Figure PQ RP QRExample vectors, show that if two triangles have corresponding sides parallel, that thelengths of corresponding sides are the sides of the two triangles byU;V;WandU0;V0;W0respectively. The hypothesis isthat there are scalarsa;b;csuch thatU0=aU;V0=bU;W0=cW. The conclusion is thata=b= show this, we start with the result of example 1; since these are the sides of a triangle, we have( )U+V+W=0;U0+V0+W0=0;or;what is the same;aU+bV+cW=0 The first equation gives usU= V W, which, when substituted in the last equation gives( )(b a)V+(c a)W=0 Now, ifb6=a, this tells us thatVandWare parallel, and so the triangle lies on a line: that is, there is notriangle.

5 Thus we must haveb=a, and by the same reasoning,c= 13 Vector Vectors in the PlaneThe advantage gained in using vectors is that they are moveable, and not tied to any particular coordinatesystem. As we have seen in the examples of the previous section, geometric facts can be easily derivedusing vectors while working in coordinates may be cumbersome. However, it is often the case, that inworking with vectors we must do calculations in a particular coordinate system . It is important to realizethat it is the worker who gets to choose the coordinates ; it is not necessarily inherent in the now explain how to move back and forth between vectors and coordinates . Suppose, then, that acoordinate system has been chosen: a pointO, the origin, and two perpendicular lines through the origin,thex- andy-axes.

6 A vectorVis determined by its length,jVjand its direction, which we can describe bythe angle thatVmakes with the horizontal (see figure ). In this figure, we have realizedVas thevector~OPfrom the origin toP. Let(a;b)be the cartesian coordinates ofP. Note thatVcan be realizedas the sum of a Vector of lengthaalong thex-axis, and a Vector of lengthbalong they-axis. We expressthis as letIrepresent the Vector from the origin to the point (1,0), andJthe Vector fromthe origin to the point (0,1). These are thebasicunit vectors (a unit Vector is a Vector of length 1). Theunit Vector in the direction iscos I+sin a Vector of lengthrand angle , thenV=r(cos I+cos J). IfVis the Vector from the originto the point(a;b);ris the length ofV, and cos I+cos Jis its direction.

7 IfP(a;b)is the endpoint ofV, thenV=~OP=aI+ called (a;b)jVjIJab Of course,rand are the usual polar coordinates , and we have these relations:( )jVj=pa2+b2; =arctanba;a=jVjcos ;b=jVjsin :We add vectors by adding their components, and multiply a Vector by a scalar by multiplying the com-ponents by the +bJandW=cI+dJ, thenV+W=(a+c)I+(b+d) is verified in figure in the Plane189 Figure +WVWabcdFigure + 25currentExample boy can paddle a canoe at 5 mph. Suppose he wants to cross a river whose currentmoving at 2 mph. At what angle to the perpendicular from one bank to the other should he direct hiscanoe?Draw a diagram so that the river is moving horizontally from left to right, and the direct crossingis vertical (see figure ). Place the origin on the lower bank of the river, and choose thex-axis inthe direction of flow, and they-axis perpendicularly across the river.

8 TIn these coordinates , the velocityvector of the current is 2I. LetVbe the velocity Vector of the canoe. We are given thatjVj=5 and wewant the resultant of the two velocities to be vertical. If is the desired angle, we see from the diagramthat sin =2=5, so =23:5 .Example object on the plane is subject to the three forcesF=2I+J;G= 8J;H. Assumingthe object doesn t move, findH. At what angle to the horizontal isHdirected?By Newton s law, the sum of the forces must be zero. Thus( )H= F G= 2I J+8J== 2I+7J:If is the angle from the positivex-axis toH, tan = 7=2, so =105:95 , sinceHpoints upwardand to the vectors represent magnitude and length, we need a computationally straightforward way ofdetermining lengths and angles, given the components of a productof two vectorsV1andV2is defined by the equation( )V1 V2=jV1jjV2jcos ;where is the angle between the two that since the cosine is an even function, it does not matter if we take fromV1toV2, or in theopposite sense.

9 In particular, we see thatV1 V2=V2 V1. Now, we see how to write the dot product interms of the components of the two +b1 JandV2=a2I+b2J. Then( )V1 V2=a1a2+b1b2 Chapter 13 Vector Algebra190with equality holding only when the vectors are see this, we use the polar representation of the vectors:( )V1=r1(cos 1I+sin 1J);V2=r2(cos 2I+sin 2J):Then( )V1 V2=r1r2cos( 1 2)=r1r2cos 1cos 2+r1r2sin 1sin 2by the addition formula for the cosine. This is the same as( )V1 V2=(r1cos 1)(r2cos 2)+(r1sin 1)(r2sin 2)which is equation ( ) in Cartesian coordinates . As for the last statement, we have strict inequalityunless cos =1, that is =0 or , in which case the vectors are ) Two vectorsVandWare orthogonal if and only ifV W= ) IfLandMare two unit vectors withL M=0, then for any vectorV, we can write( )V=aL+bM;witha=V L;b=V M;andjVj=pa2+b2:We shall say that a pair of unit vectorsL;MwithL M=0 form abasefor the plane.

10 This statementjust reiterates that we can put cartesian coordinates on the plane with any point as origin and coordinateaxes two orthogonal lines through the origin; that is the lines in the directions ofLandM. To show partb) we start with figure that figure, we see that we can write any Vector as a sumV=aL+bMwith (by the Pythagoreantheorem)jVj=pa2+b2. We now show thata;bare as described;( )V L=(aL+bM) L=aL L+bM L=a:SimilarlyV M= the angle between the vectorsV=2I 3 JandW=I+ havejVj=p22+32=p13;jWj=p12+22=p5 andV W=2(1)+( 3)(2)= 4. Thus( )cos =V WjVjjWj= 4p65= in the Plane191so = 119:7 .Example we have put cartesian coordinates on the plane, withI;Jthe standard base. Let( )L=I+Jp2;M= I+Jp2be a new base. Given the pointP(5;2), write~OPin terms the preceding proposition,( )~OP L=(5I+2J) I+Jp2 =7p2;~OP M=(5I+2J) I+Jp2 = 3p2;so~OP=(7L 3M)= , using vectors, that the interior angles of an isosceles triangle are +W 0In figure we have labelled the sides of equal length asVandW.


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