VectorsPreCalculus Presentation by: History - John Klein pg. 2 Review - Brad Parbs pg. 3 Solving a problem - Sam Weber pg. 4 Applications - Mike Zabrowski pg. 5 Works Cited pg. 6

avaiable online at BradParbs.com/vectors

VectorswerefirstthoughtaboutbyIsaacNewtoninhisbook,PrincipiaMathematicawrit-tenin1687.InthePrincipiaMathematica,Newtondealtextensivelywithwhatarenowconsid-eredvectorialentities,likevelocityandforce,butnevertheconceptofavector.Thesystematicstudyanduseofvectorswerea19thandearly20thcenturyphenomenon.Vectorswereborninthefirsttwodecadesofthe19thcenturywiththegeometricrepresentationsofcomplexnum-bers.CasparWessel,JeanRobertArgand,andCarlFriedrichGaussallfirstconceivedtheideaofcomplexnumbersaspointsinthetwo-dimensionalplane.Mathematiciansandscientistsworkedwithandappliedthesenewnumbersinvariousways;forexample,GaussmadecrucialuseofcomplexnumberstoprovetheFundamentalTheoremofAlgebra.In1837andWilliamRowanHamiltonshowedthatthecomplexnumberscouldbeconsideredabstractlyasorderedpairs(a,b)ofrealnumbers.Thisideawasapartofthecampaignofmanymathematicians,includingHam-iltonhimself,tosearchforawaytoextendthetwo-dimensional“numbers”tothreedimensions;butnoonewasabletoaccomplishthis,whilepreservingthebasicalgebraicpropertiesofrealandcomplexnumbers. Afteragooddealoffrustration,in1843Hamiltonwasfinallyinspiredtogiveupthesearchforsuchathree-dimensionalnumbersystemandinsteadheinventedafour-dimensionalsystemthathecalledquaternions.Hamiltonhadbeenknightedin1835,andhewasawell-knownscientistwhohaddonefundamentalworkinopticsandtheoreticalphysicsbythetimeheinventedquaternions,sotheyweregivenimmediaterecognition.Inturn,hedevotedtheremain-ing22yearsofhislifetotheirdevelopmentandpromotion.Hewrotetwoexhaustivebooks,LecturesonQuaternions(1853)andElementsofQuaternions(1866),detailingnotjustthealgebraofquaternionsbutalsohowtheycouldbeusedingeometry.Atonepoint,Hamiltonwrote,“Istillmustassertthatthisdiscoveryappearstometobeasimportantforthemiddleofthenine-teenthcenturyasthediscoveryoffluxionswasforthecloseoftheseventeenth.”Heacquiredadisciple,PeterGuthrieTait(1831--1901),whointhe1850sbeganapplyingquaternionstoprob-lemsinelectricityandmagnetismandtootherproblemsinphysics.Inthesecondhalfofthe19thcentury,Tait’sadvocacyofquaternionsproducedstrongreactions,bothpositiveandnegative,inthescientificcommunity.(Vectors)AtaboutthesametimethatHamiltondiscoveredquaternions,HermannGrassmannwroteTheCalculusofExtension.Inthisbookhestudiedthetheoryoftides.Grassmannexpand-edtheconceptionofvectorstoanarbitrarynumber.Then,Grassmannreceivedverylittlecreditforhisdiscoveryduetohisobscurestyleandbecausehewasaschoolteacherwithoutamajorscientificreputation. ThedevelopmentofthealgebraofvectorsandofvectoranalysisasweknowittodaywasfirstrevealedinsetsofremarkablenotesmadebyJ.WillardGibbsforhisstudentsatYaleUni-versity.Heconcludedthatvectorswouldprovideamoreefficienttoolforhisworkinphysics.So,beginningin1881,Gibbsprivatelyprintednotesonvectoranalysisforhisstudents,whichwerewidelydistributedtoscholarsintheUnitedStates,Britain,andEurope.ThefirstbookonmodernvectoranalysisinEnglishwasVectorAnalysisin1901.Gibbs’snoteswereassembledbyoneofhislastgraduatestudents,EdwinB.Wilson.Vectorsarenowthemodernlanguageofagreatdealofphysicsandappliedmathematicsandtheycontinuetoholdtheirownintrinsicmathematical interest.

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Throughout the study of Pre Calculus, vectors are studied for quite some time. Also, in Physics, an understanding of vectors is essential. Basically, there are a few different attributes and features of vectors that need to be understood, then you can move onto more complex problems involving vectors. Many problems in Physics as well as Pre Calc are solved using vectors. We write a vector by writing the initial point as a letter, then the end point also as a letter, then draw a half arrow on top. To show magnitude, we write the magnitude, then surround it with two lines on both sides. The magnitude is also the length. Vectors are very similar to line segments when written, and we need to take care not to confuse them. There are two things to remember about a vector, it is made up of a size and a direction. The only difference between different vectors is going to be the size or the direction. When a vector is at an angle, one can easily break it down into two components, both parallel to either the x or y axis. When we break it down, we need to make sure to look at the angles and sizes, to make sure no mistakes are made. If we break the vector down, and we end up with a vector having a magnitude of one, that vector is a unit vector, used only for direction. This component form, when broken down, can also be written out in brackets. We put the vector at (0,0) and we write the end point in brackets. <3,3> or <5,2>. This allows us to easily do mathematics and solve for different things. To aid in solving problems, we use unit vectors. Unit vectors have a magnitude, that is, the size, of one. Magnitude is basically the power of it. Unit vectors help to indicate direction. A key concept when using vectors, is the mathematics involved. In the past, we have learned addition and scalar multiplication. Addition is quite simple, we simply place the two vectors together, and draw in a third vector from the starting point of the first vector to the end point of the second vector. The size and direction of this resulting vector is the answer, or simply the resultant. Scalar multiplication is also quite simple, once you know how to. Scaling a vector is multiplying it by a certain factor. We take the vector in the bracketed form, then multiply each number by the scale. A vector going to <2,2> scaled by 2 would end up at <4,4>. We can also break a vector into a linear combination form. This allows us to use the horizontal and vertical components easier. We take the first number, put that in front of i, the second number in front of j. We can easily add vectors in linear form, as well as finding the angle between them. To find the angle, we take the tangent of the first number of the first vector over the first number of the second vector. We also take the tangent of the second numbers, then add those both together. The dot product is also used often, and is found by taking the magnitude of two vectors, multiplying them together, also multiplying the cos of the angle between them.

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F1= 65N at 70°F2= 55N at 20°

1) Draw the components out and figure out sin x and sin y. then find out cos x and cos y. By drawing the components we find out which values we need to find when using sin and cos. For sin we need to opposite and hypotenuse side. For cos we need the adjacent side and hypotenuse side. By finding the values we then know what we are solving for.

2) Once you draw the components and find out both sin and cos of x and y you need to find their sums. You find the values of the numbers and then add. You need to multiply the number in Newtons by cos of the degree of the vector.

3) After adding your sin and cos values you have found the components of the resultant. The components of the resultant are used to find the magnitude.

The resultant <80,74>

4) To find the magnitude, which is the size of the vector, you need to square each component of the resultant and then find the square root of that sum.

√ 802 + 742 = 108.98

5) Finding the magnitude resembles using the Pythagorean Theorem. You can also find components if the magnitude is given along with another component.

For example if the given magnitude is 108.98 and 80 as the component. You start by squaring the magnitude. 108.982

Then you get 11876.6404.

Then you square the component 80.

Then you get 6400 Then you subtract the magnitude squared and the component squared. 11876.6404-6400 = 5476.6404 Then you need to find the square root of the difference. √ 5476.6404After those steps you come back to 74, which was the other component.

55 cos (20)+ 65 cos (70) + 74

55 sin (20)+ 65 sin (70)+ 80

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Asweknow,avectorisanythingwithasizeandadirection.Assuch,therearemanyexamplesofvectorsineverydaylife.First,lootatamovingcar.Acarmovesatacertainspeed(inthiscase,thesize)andmovesinadirection.Therefore,acartravellingdown a road is a vector. Therearemanyotherexamplesofvectorsintheworldofsports.Forexample,whenasoccerballiskickeditusuallymovesinastraightpath—size(speed)anddirection.Withthebootofaskilledsoccerplayer,however,thiskickcanbemadeintoacurve.Allofthedifferentforcesactingontheball--suchastheforceofthekickthatisbroughtonbythepersonandthespinoftheballintheair--deflectstheball’spathandresultsinacurvedflightpattern. Theconceptofaballmovingataspecificvelocitythroughtheairorgroundiscommoninmanyothersportsaswell.Otherexamplesinclude:afootballthrownbyBrettFavre,acurveballthrownbyanMLBpitcher,aslapshotfromahockeyplayeronapuck,andevenabowlingballthrowndownanalley. Also,vectorscanbeseenwithinotherthingsapartfromsports.Theycanalsobeseenintheworldofcomputers.Whenapersonwantstotellthecomputertomakeanimage,hecoulddosobyusingpixels.Pixelscouldbeusedbytellingthecomputerexactlywhatcoloreverysingleindividualpixelis.However,thisisnotthebestwaytodothis.Makingavisualbythismethodtakesupalotofmemory,andwhentheangleismovedthepicturegetsdistorted.Itismucheasiertousevectorswhencreatingthisimage. Tousevectors,onecantellthecomputerwhatdirectionsunlightorwindiscomingfrom.Bothsunlightandwindhaveadirectionandasizeofhowstrongthewindgustisorhowintensethesunisbeatingdown.Whenusedlikethis,thecolorofthepicturecanbecalculatedbythecomputertogiveitrealisticlightingandwindeffects. Therearealsomanyotherusesthatavectorhas.Forexample,meteorologistsusevectorstopredictweatherpatterns.Theyusethewindspeedandthedirectionthatitistravellingtogivearoughestimateofhowtheweatherwillbeinacertainarea.Vectorsalsocanbeusedwhensailinginasailboat.Thewindpushestheboatwithaforceandthe boat moves in a certain direction. Vectorsaresosimplethattheycanbefoundinmanyaspectsoflife.Onceagain,anythingwithsizeanddirectionisavector,soanymovingobjectcanbeavector.

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6Works Cited

“Basic Vector Operations.” Basic Vector Operations. 18 May 2009 <http://hyperphysics.phy-astr.gsu.edu/hbase/

vect.html>.

“Vector -- from Wolfram MathWorld.” Wolfram MathWorld: The Web’s Most Extensive Mathematics Resource.

21 May 2009 <http://mathworld.wolfram.com/Vector.html>.

“Vectors: Fun activities with vectors.” Division of Engineering Welcome. 27 May 2009 <http://www.engin.

brown.edu/courses/en3/Notes/Vector_Web2/vectors7/vectors7.htm>.

“Vectors.” Mathematics and Statistics | The Department of Mathematics and Statistics. 27 May 2009 <http://

www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html>.

“Vectors.” The Math Forum @ Drexel University. 22 May 2009 <http://mathforum.org/~klotz/Vectors/vectors.

html>.

“Vectors.” The Physics Department - Mechanics, Vectors. 17 May 2009 <http://id.mind.net/~zona/mstm/

physics/mechanics/vectors/vectors.html>.