VectorsWhat is a Vector?Vector NotationSlide 4Equivalent VectorsFind the VectorFundamental Vector OperationsVector AdditionVector SubtractionVector Addition / SubtractionMagnitude of a VectorUnit VectorsSlide 13Finding the ComponentsAssignment Part AApplications of VectorsApplication of VectorsDot ProductFind the Dot (product)Dot Product FormulaFind the AngleDot Product Properties (pg 321)Assignment BScalar ProjectionSlide 25Parallel and Perpendicular VectorsWork: An Application of the Dot ProductAssignment CVectorsLesson 4.32What is a Vector?A quantity that has bothSizeDirectionExamplesWindBoat or aircraft travelForces in physicsGeometricallyA directed line segmentInitial pointTerminal point3Vector NotationGiven byAngle brackets <a, b> a vector with Initial point at (0,0)Terminal point at (a, b)Ordered pair (a, b)As above, initial point at origin, terminal point at the specified ordered pair(a, b)4Vector NotationAn arrow over a letteror a letter in bold face VAn arrow over two lettersThe initial and terminal points or both letters in bold face ABThe magnitude (length) of a vector is notated with double vertical linesVurVABABuuur5Equivalent VectorsHave both same direction and same magnitudeGiven pointsThe components of a vector Ordered pair of terminal point with initial point at (0,0) (a, b)( ) ( ), ,t t t i i iP x y P x y,t i t ix x y y- -6Find the VectorGiven P1 (0, -3) and P2 (1, 5)Show vector representation in <x, y> format for <1 – 0, 5 – (-3)> = <1,8>Try theseP1(4,2) and P2 (-3, -3)P4(3, -2) and P2(3, 0) 1 2PPuuuur7Fundamental Vector OperationsGiven vectors V = <a, b>, W = <c, d>MagnitudeAddition V + W = <a + c, b + d>Scalar multiplication – changes the magnitude, not the direction3V = <3a, 3b> 2 2V a b= +8Vector AdditionSum of two vectors is the single equivalent vector which has same effect as application of the two vectorsABA + BNote that the sum of two vectors is the diagonal of the resulting parallelogramNote that the sum of two vectors is the diagonal of the resulting parallelogram9Vector SubtractionThe difference of two vectors is the result of adding a negative vectorA – B = A + (-B)AB-BA - B10Vector Addition / SubtractionAdd vectors by adding respective components<3, 4> + <6, -5> = ?<2.4, - 7> - <2, 6.8> = ?Try these visually, draw the resultsA + CB – AC + 2BABC11Magnitude of a VectorMagnitude found using Pythagorean theorem or distance formulaGiven A = <4, -7>Find the magnitude of these:P1(4,2) and P2 (-3, -3)P4(3, -2) and P2(3, 0) 2 24 ( 7)A = + - =12Unit VectorsDefinition:A vector whose magnitude is 1Typically we use the horizontal and vertical unit vectors i and ji = <1, 0> j = <0, 1>Then use the vector components to express the vector as a sumV = <3,5> = 3i + 5j13Unit VectorsUse unit vectors to add vectors<4, -2> + <6, 9>4i – 2j + 6i + 9j = 10i + 7jUse to find magnitude|| -3i + 4j || = ((-3)2 + 42)1/2 = 5Use to find directionDirection for -2i + 2j( )2tan234qpq-==14Finding the ComponentsGiven direction θ and magnitude ||V||V = <a, b>6V =bacossina Vb Vqq= �= �6pq =15Assignment Part ALesson 4.3APage 325Exercises 1 – 35 odd16Applications of VectorsSammy Squirrel is steering his boat at a heading of 327° at 18mph. The current is flowing at 4mph at a heading of 60°. Find Sammy's courseNote info about E6B flight calculatorNote info about E6B flight calculator17Application of VectorsA 120 pound force keeps an 800 pound box from sliding down an inclined ramp. What is the angle of the ramp?What we haveis the forcethe weightcreatesparallel to theramp18Dot ProductGiven vectors V = <a, b>, W = <c, d>Dot product defined asNote that the result is a scalarAlso known asInner product orScalar productV W a c b d= �+ �g19Find the Dot (product)Given A = 3i + 7j, B = -2i + 4j, and C = 6i - 5j Find the following:A • B = ?B • C = ?The dot product can also be found with the following formulacosV W V W a= � �g20Dot Product FormulaFormula on previous slide may be more useful for finding the angle coscosV W V WV WV Waa= � �=�gg21Find the AngleGiven two vectorsV = <1, -5> and W = <-2, 3>Find the angle between themCalculate dot productThen magnitudeThen applyformulaTake arccosVW22Dot Product Properties (pg 321)CommutativeDistributive over additionScalar multiplication same over dot product before or after dot product multiplicationDot product of vector with itselfMultiplicative property of zeroDot products ofi • i =1j • j = 1 i • j = 023Assignment BLesson 4.3BPage 325Exercises 37 – 61 odd24Scalar ProjectionGiven two vectors v and wProjwv = vwprojwvThe projection of v on wcosv a�25Scalar ProjectionThe other possible configuration for the projectionFormula used is the same but result will be negative because  > 90°vwprojwvThe projection of v on wcosv a�26Parallel and Perpendicular VectorsRecall formulaWhat would it mean if this resulted in a value of 0??What angle has a cosine of 0?cosV WV Wa�=�0 90V WV Wa�= � =�o27Work: An Application of the Dot ProductThe horse pulls for 1000ft with a force of 250 lbs at an angle of 37° with the ground. The amount of work done is force times displacement. This can be given with the dot product37°W F s= �cosF s a= � �28Assignment CLesson 4.3CPage 326Exercises 63 - 77 odd79 – 82