Vectors and Plane Geometry - math.hawaii.edu

1 Vectors and Plane GeometryKarl Heinz DovermannProfessor of MathematicsUniversity of HawaiiJanuary 27, 2011 PrefaceDuring the first week of the semester it is difficult to get started with thecourse material. Some students have not settled in, some arestill changingsections, and some still have to sign up for a course. For thisreason, it isreasonable to teach an interesting, relevant topic that is somewhat indepen-dent of the course. Some instructors in a calculus course usethe first weekto review topics from precalculus. Instead, we decided to spend this weekon Vectors and the Geometry of the Plane , topics that other sciences andengineering like to see covered early. These notes are meantas lecture notesfor a one-week is nothing original in these notes. The material can befoundin many places. Many calculus books will have a section on Vectors in thesecond half, but students would not like to start reading there.

2 The materialis also contained in a variety of other mathematics books, but then we wouldnot want to force students to acquire another book. For thesereasons, weare providing these 1 The Algebra of VectorsSome information is completely described by a single number, such as thebalance of your checking account at a specific moment. But if you like torecord the motion of a billiard ball on a pool table, then you will need threenumbers. You need to record the position of the ball in a planeat any giventime. An array whose entries are real numbers is an example ofa vector , nomatter how many entries the array may have. We may add vectorsand wemay multiply them by numbers, and the rules for this arithmetic offer Definition of a VectorP=H2,3L-11234x-11234yFigure : vector inR2 PPPPPP-11234x-11234yFigure : Same vectors12 CHAPTER 1.

3 THE ALGEBRA OF VECTORSWe will content ourselves withvectorsin the cartesian planeR2or inthree dimensional spaceR3. PointsPin the Plane are described by pairs(a, b) of real numbers, whereaandbstand for thexandycoordinates ofthe pointP. This means, when we projectPperpendicularly on thex axis,then we geta, and when we projectPon they axis, then we getb, seeFigure occasions, we identify the point or vectorPwith an arrow from theorigin (0,0) of the coordinate system toP. We may even denote both objectswith the same symbol. In this description the Vectors becomemovable. Theyare characterized by their direction and length. In Figure you see thevectorPmoved (by parallel translations) to different positions in the of the arrows still vectorufromQ= (a1, b1) toP= (a2, b2) can be written as( )u= QP= (a2 a1, b2 b1).Its tail isQand its tip isP.

4 For and illustration, see Figure :u= QPbaHa,bLa2+b2-11234x-11234yFigure : LengthExample vector fromQ= (1,2) toP= (5,3) is QP= (5 1,3 2) = (4,1).Exercise the vectoruwith tail(1,2)and head(2, 1). In thesame set of coordinates, sketch the vectoruwith(3,1)as its tail, and with(4,2)as its ADDITION AND MULTIPLICATION WITH SCALARS3 The two ways of viewing Vectors , points in the Plane versus arrows, arerelated by the formulaP= OPwhereO= (0,0) is the origin of the coordinate system. Both, the point andthe arrow, are shown in Figure three dimensional spaceR3we have three coordinate axes, often calledthex,y, andz axes. The first two are used in a horizontalx-y Plane , andthez axis is perpendicular to thex-y Plane pointing upwards. Accordingly, Vectors inR3are triples of real length||u||of a vectoru= (a, b), illustrated in Figure , is( )||u||=pa2+ is the Euclidean distance between the points (0,0) and (a, b), or the lengthof the line segment that joins these two length of the vectoru= (2,3) is||u||=p22+ 32= length of the vector fromQ= (1,2) toP= (5,3) is|| QP||=||(4,1)||=p42+ 12= may also write Vectors as columns, say (34).

5 This is more naturalin some contexts. Row Vectors take up less space in type setting. For mostpurposes, the difference is only notational, not the length of the vectorsu= (1,4),v= (1,4,2)andw= 5 2 . Find the length of the vector QPfromQ= (1,5)toP= (3,2). Addition and Multiplication with ScalarsWe consider two operations. Ifv= (x1, y1) andw= (x2, y2) are Vectors ,then we define their sum to be the vector :( )v+w= (x1+x2, y1+y2)Ifcis a real number andv= (x, y), then we define the scalar product ofcandvto be the vector ( )cv=c(x, y) = (cx, cy).4 CHAPTER 1. THE ALGEBRA OF VECTORSIn a more general setting,ccould be taken from a different set of numbers,not the real numbers. To allow this, it is common to callca scalar. For us,a real number and a scalar are the might indicate the multiplication by a dot, and writec vinstead ofcv, but this is only rarely done.

6 It is convenient to writev/cinstead the obvious reasons, we say that Vectors are added, or multiplied with ascalar, coordinatewise. The operations can be applied alsoto Vectors inR3,or Vectors with any number of (1,2) + (5,2) = (6,4) & 3(2,7) = (6,21).For column Vectors with three coordinates we haveExample 132 + 3 27 = 419 & 4 2 15 = 8 420 Exercise (1,2)andv= (2, 3).1. Find2u,u+v,u+ 2v, andv FindPif QP= 2u+vandQ= ( 1, 1).3. FindQif QP=u vandP= (5,2).In the planeR2it is common to set( )i= (1,0) andj= (0,1),and in 3-dimensional space( )i= (1,0,0),j= (0,1,0) andk= (0,0,1).Using this notation,(2, 3) = 2i 3jand 2 35 = 2i 3j+ is common to refer toiandjas standard basis Vectors ofR2, or to{i,j,k}as the standard basis ADDITION AND MULTIPLICATION WITH SCALARS5 Exercise (9,5)in terms of the standard basis arithmetic of Vectors offers no surprise:Proposition thatu,v, andware any Vectors inR2andcanddare scalars, +wandcvare + (v+w) = (u+v) + +w=w+ There exists a vector0so thatv+ 0 = 0 +v= There exists a vectorv inR2so thatv+v = (v+w) =cv+ (c+d)v=cv+ (cd)v=c(dv).

7 In (1) we are asserting that the sum of Vectors is a vector , andso is ascalar multiple of a vector . In (2) we are saying that it does not matterin which order the additions are carried out, the result is the same. Thisproperty is called the associative law. In (3) we are saying that the additionof Vectors is commutative, we may interchange the summands and the resultis (4) we assert that there is a zero for the addition of Vectors . Addingzero does not change a vector . Such an element is also is called a neutralelement for addition, and it is unique. Obviously, 0 = (0,0) is the vectorboth of whose coordinates are words, (5) says that every vectorvhas an additive inversev . Nec-essarily, and also in a more general setting, it will be unique. Ifv= (a, b),thenv = ( a, b). It is common to denote the additive inverse ofvby is consistent with common arithmetic to setu v=u+ ( v),and call this operation remaining three properties are called distributive 1.

8 THE ALGEBRA OF VECTORSR emark and multiplication of Vectors inR3obeys the samelaws as the ones spelled out in Proposition One may also consider asetVwith two operations, like the addition and scalar multiplication fromabove, and call it a real vector space if properties (1) (8) in Proposition One important example of such a vector space is the set of all realvalued functions that are defined on some domain. One adds such functionsby adding the values, and one multiplies then with scalars bymultiplyingthe values with scalars. This algebra on functions is one of the importanttopics of a precalculus Geometric Interpretation of vector operations have geometric interpretations. Ifuandvare vectorsin the Plane , thought of as arrows with tips and tails, then wecan constructthe sumw=u+vas shown in Figure We arrange it so that the tip ofuis the tail ofv.

9 Thenwis the vector whose tail is the tail ofuand whosetip is the tip ofv. In other words we concatenate the two +v12345x123yFigure : vector addition-u2uuu 2-11234x-11234yFigure : Scalar multiplesSupposeuis a vector andca scalar. Ifcis positive, thencuhas the samedirection asuandctimes its length, so||cu||=c||u||. Ifcis negative, thencuhas length|c| ||u||, and its direction is opposite to that ofu. Examplesare shown in Figure ( 2,1)andv= (3,2), GEOMETRIC INTERPRETATION OF Sketchuandvas arrows with tail at(1,1).2. Construct the sum Sketch2uas an arrow with tail at(1, 2).4. Sketch uas an arrow with tail at(1, 1).In physics Vectors are often used to describe forces, and forces add asvectors do. This principle is applied in the following water balloon with a mass of 10kg (containing about twoand a half gallons of water) hangs from two wires (WL to the left and WR tothe right) as shown in Figure It is suspended from the pointC= (3,2)Calculate the tension (force) in the wires that results fromthe gravitationalpull on the ,4LB=H5,3 LWLWRC=H3,2L-112345x-112345yFigure : Water balloonLet us clarify the units for our computation.

10 If a force of 1 Newton actson a mass of 1kg, then the resulting acceleration is 1m/sec2. Gravitationwould result in an acceleration of Thus, the gravitational forceon the balloon will be 98 the point where the three wires are joined. We suppose thatC= (3,2). Three forces act onC, the vertical forceFresulting from the8 CHAPTER 1. THE ALGEBRA OF Vectors gravitational pull on the water balloon, the tensionT Lin the wire to theleft, and the tensionT Rin the wire to the right. The three forces canceleach other out, as the system is at rest:( )F+T L+T R= know thatF= (0, 98). The forces (tension) in the wires break upinto a horizontal and vertical component:T L= (T Lh, T Lv) andT R= (T Rh, T Rv).We use the Geometry of the situation to calculate the components. Let be the angle between the positive horizontal axis (through the pointCand pointing to the right) and the wire WR.