Applied_Vectors_Vectors2 Notes

GeneralTips: ● Drawingadiagramhelpsmassively.Regardlessofwhetherit’sabstractvectorsorapplication.● Practicethestandardmethodsandgetgoodatthem(Footofperpendicular,lineofintersection,findinganglesbetweenvectors,etc.)● Befamiliarwiththedifferentformsoflinesandplanes(Cartesian, Parametric,Vector, Scalar product)● Understandthegeometricsignificanceofthedot/crossproducts○ Dot:Testforperpendicular,lengthofprojection,angle,etc.○ Cross:Areasweptout,normalvector,lengthofoppositeetc. AsummaryofBasicFormulae: Products:𝑎 • 𝑏 = |𝑎||𝑏|𝑐𝑜𝑠(Θ) = 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 (𝑖𝑛 3𝐷),whereistheunitvectorperpendiculartobothvectors,whose𝑎 × 𝑏 = |𝑎||𝑏|𝑠𝑖𝑛(Θ)𝑛 𝑛directionisgivenbytherighthandrule.(Component-wiseformulagiveninMF26,althoughifI’mnotmistaken,theyremoveditfromMF27.) Inparticular:If and areperpendicular,𝑎 𝑏 𝑎 • 𝑏 = 0If and areparallel,𝑎 𝑏 𝑎 × 𝑏 = 0givestheareaoftheparallelogramwhosesidesare and.|𝑎 × 𝑏| 𝑎 𝑏 Equation(s)ofaline:,whereisthepositionvectorofapointontheline,and isthedirection𝑟 = 𝑎 + λ𝑑 𝑎 𝑑vectoroftheline.(Alineisdefinedbyapointandadirection.) By:u/A_Strolling_Orca 1

Bywritingtheaboveequationoutcomponent-wiseandequatingineach,weget:λ ,providedthatnoneof arezero.Ifoneis0, 𝑥−𝑎1𝑑1 = 𝑦−𝑎2𝑑2 = 𝑧−𝑎3𝑑3 𝑑1,𝑑2,𝑑3 onethecoordinatesisfixed. E.g.𝑟 = (2,1,0) + λ(1,1,0) ↔ 𝑥 − 2 = 𝑦 − 1, 𝑧 = 0. Equation(s)ofaplane:,whereisthenormalvectortotheplane,and isthepositionvectorof𝑟 • 𝑛 = 𝑎 • 𝑛 𝑛 𝑎agivenpointintheplane.(Aplaneisdefinedbyapointandanormal).WecanalsogettheCartesianequationbywriting andevaluatingtheright𝑟 = (𝑥,𝑦,𝑧), 𝑛 = (𝑛1,𝑛2,𝑛3)handside.Notethatif isaunitvector, representsthelengthofprojectionof𝑛 𝑎 • 𝑛 𝑎onto,whichistheperpendiculardistancefromtheorigintotheplane.𝑛 Aplanecanalsobedefinedintermsof2linearlyindependent(i.e.notparallel)vectorsparalleltoit,orequivalently,threenon-collinearpointsitcontains.Thedifferencevectorsbetweenthe3pointsgiveyouthe2vectorsitcontains. ,whereisapointintheplane,and arevectorsparalleltothe𝑟 = 𝑎 + λ𝑏 + µ𝑐 𝑎 𝑏 𝑐plane,with foranyrealscalar, and, realparameters.𝑏 ≠ 𝑘𝑐 𝑘 λµ E.g. definesaplane.𝑟 = (1,1,1) + λ(1,0,0) + µ(0,1,0) doesnotdefineaplane,asthetwo𝑟 = (1,2,3) + λ(1,0,0) + µ(− 1,0,0)vectorsgivenareparallel. Takingthecrossproductofthetwovectorsgivesyouthenormaltotheplane,therebylettingyouconvertfromthisparametricformtothecartesianform.Convertingfromthecartesiantotheparametricissimplyamatteroffinding3non-collinearpointsontheplane,butIhaveneverseenaquestionaskforthis. Angles:Foranytwovectors ,theangle betweenthemisgivenby:𝑎,𝑏 Θ𝑐𝑜𝑠(Θ) = (𝑎•𝑏)|𝑎||𝑏| By:u/A_Strolling_Orca 2

Foranytwoplaneswithnormalvectors theacuteanglebetweentheplanesis𝑛1,𝑛2,givenby: 𝑐𝑜𝑠(Θ) = |(𝑛1•𝑛2)||𝑛1||𝑛2| Foralinewithdirectionvector intersectingaplanewithnormal,theangleit𝑑1 𝑛1makeswiththeplaneisgivenby: 𝑠𝑖𝑛(Θ) = |(𝑛1•𝑑1)||𝑛1||𝑑1| Distances/Lengths:Fortwoparallelplanes, and ,thedistancebetweenthemisgiven𝑟 • 𝑛 = 𝑑1 𝑟 • 𝑛 = 𝑑2by: .𝐷 = |𝑑1 − 𝑑2|/|𝑛|(Thisformulacanbederivedbyconsideringtheprojectionofpositionvectorsofpointsonthetwoplanesontotheircommonnormal.) Lengthofprojecti