Maths 21a First Midterm ReviewOliver Knill, July 12, 2011Lines and Planes Parametrized curves Surfaces QuadricsParametric SurfacesOther coordinates Distance formulasPlan: Tuesday, July 12, 2011Points and VectorsP = (3,0,1) Q = (2,5,7) The components of vare the differences between the coordinates of Q and P.Tuesday, July 12, 2011Points and VectorsP = (3,0,1) Q = (2,5,7) v = PQ = < -1,5,6> The components of vare the differences between the coordinates of Q and P.Tuesday, July 12, 2011Additionv = < -2,5,6> w = < 1,2,3> v +w= < -1,7,9> Tuesday, July 12, 2011Subtractionv = < -2,5,6> w = < 1,2,3> v -w= < -3,3,3> Tuesday, July 12, 2011Distancesv = < -2,2,1> PQd(P,Q) = | v |Tuesday, July 12, 2011SpheresQ=(3,-4,2)r=5(x-3) + (y+4) + (z-2) = 25 2 2 2PcenterTuesday, July 12, 2011Dot and Cross product = 6-4+1 = 3x =i j k3 4 12 -1 2 = <9,-4,-11>v = < 3,4,1> w = < 2,-1,1>vwvwvwTuesday, July 12, 2011Two important formulasvwv=wcos(α )vwxv=wsin(α)Tuesday, July 12, 2011Projectionv=< 2,-3,4 > w = <4,0,1>Project v onto w: wproj (v)= v www.= 12 17 <4,0,1>wscalar projection = componentTuesday, July 12, 2011Projectionv=< 2,-3,4 > w = <4,0,1>Project v onto w: wproj (v)= v www.= 12 17 <4,0,1>wscalar projection = componentTuesday, July 12, 2011Lines and PlanesP=(3,4,1)OP = <3,4,1>vwv = <1,1,-3>w = <1,-2,1>r(t,s) = < 3+t+s,4+t-2s,1-3t+s>Parametrizationnn = <7,-4,-3>7x-4y-3z=2Tuesday, July 12, 2011Linesv = <2,1,-3>P=(3,4,1)Parametrizationr(t) = < 3+2t,4+t,1-3t>(solve for t)= < x,y,z>x-32=y-41z-1=-3SymmetricequationsTuesday, July 12, 2011ProblemFind the equation of the plane passing through the points A=(0,1,1),B=(2,2,2),C=(5,5,4) the symmetric equation of the normal line through A and the area of the triangle ABC. Tuesday, July 12, 2011ProblemFind the line of intersection of the plane computed before and the line plane x+y+z=1 Tuesday, July 12, 2011Distance Point/LinevαQP|QP| sin(α)QPTuesday, July 12, 2011Distance Point/LinevαQP|QP| sin(α)QPTuesday, July 12, 2011Distance Point/LinevαQP|QP| sin(α)QP|v||v|Tuesday, July 12, 2011Distance Point/LinevαQP|QP| sin(α)QP=QP x v|||v||v||v|Tuesday, July 12, 2011Distance Point-PlanenαQP|PQ| cos(α)|n||n|QP=PQ n|||n|.dd=Tuesday, July 12, 2011Distance Point-PlanenαQP|PQ| cos(α)|n||n|QP=PQ n|||n|.dd=Tuesday, July 12, 2011|n|Distance Line/LinevPQ|PQ| cos(α)|n|PQ=QP v x w|||v x w|wnα...Tuesday, July 12, 2011|n|Distance Line/LinevPQ|PQ| cos(α)|n|PQ=QP v x w|||v x w|wnα...Tuesday, July 12, 2011ProblemStarship “enterprise” officer Spock, beams from (1,1,1) to (3,4,5). The clingons can modify everything in distance 1 from the x axes. Will Spock be safe? (1,1,1)(3,4,5)xTuesday, July 12, 2011Area and Volumewuuv= < 1,2,3> v= < 3,1,4> w= < 1,1,1> The volume of the parallelepipedspanned by u,v,w isu (v x w) = (u x v) w. .||| |Tuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Line againQP x v|||v|= areabase= heightPQTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Point-Plane againPQPQ v x w| ||v x w|.vw=volumearea=heightQPTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Distance Line-LinewvQPQP v x w|||v x w|.= VolumeAreaTuesday, July 12, 2011Parametrized CurvesTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = < x(t),y(t),z(t)>r’(t) = < x(t),y(t),z(t)>velocity‘ ‘ ‘r’’(t)accelerationTuesday, July 12, 2011r(a) r(b)r(t)r(t) = <
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