7/8/2010 FIRST HOURLY Maths 21a, O.Knill, Summer 2010Name:• Start by writing your name in the above box.• Try to answer each questi on on the same page as the question is asked. If need ed , usethe back or the next empty page for work. If you need additional paper, write your nameon it.• Do not detach pages from this exam packet or unstaple the packet.• Please write neatly. Answers which are illegible for the grader can not be given credit.• No notes, books, calculators, computers, or other electronic aids can be allowed.• You have exactly 90 minutes t o compl et e your work.1 202 103 104 105 106 107 108 109 10Total: 1001Problem 1) ( 20 po ints) No justifications are needed.1)T FThe len gt h of the vector h1, 1, 1i is equal to 3.2)T FAny two distinct points A, B in space determine a unique line which containsthese two points.3)T FFor any two non-intersecting lines L, K, there is exactly one point P whichhas equ a l distance to both lines.4)T FThe graph of f(x, y) is a surface in space which is equal to the le vel surfaceg(x, y, z) = 0 of g(x, y, z) = f(x, y) − z.5)T FThe graph of the functio n f(x, y) = x2−y2is called an elliptic paraboloid.6)T FThe equ at i on ρ cos(θ) = 1 in spherical coordi n a t es defines a plane.7)T FThe vector h1, 2, 3i i s perpendicular to the plane x + 2y + 3z = 4.8)T FThe cros s product between the vectors h1, 2, 3i and h1, 1, 1i is 6.9)T FThe two parametrized curves ~r(t) = ht, t2, t6i, 0 ≤ t ≤ 1 and~R(t) =ht2, t4, t6i, 0 ≤ t ≤ 1 have the same arc length.10)T FThe point ( 1, 0, 1) has the spherical coordinates (ρ, θ, φ) = (√2, 0, π/4).11)T FThe distance between two parallel p l a n es is the distan ce of any point on oneplane to th e other plane.12)T FProj~w(~v × ~w) =~0 holds for all nonzero vectors ~v, ~w.13)T FThe vector projection of h2, 3, 4i onto h1, 0, 0i is h2, 0, 0i.14)T FThe triple scalar product ~u ·(~v × ~w) between three vectors ~u,~v, ~w is zero i fand on l y if two or more of the 3 vectors are parallel.15)T FThere are two vectors ~v and ~w so that the dot product ~v · ~w is equal to thelength of th e cross produ ct |~v × ~w|.16)T FTwo cylinders of radius 1 whose axes are lines L, K have dist a n ce d(L, K) =3 have distance 2.17)T FThere are two unit vectors ~v and ~w tha t are both par al l el and perpendicular.18)T FAssuming the curvature to exi st for all time, the curvature κ(~r(t)) is alwayssmaller than or equ al to |~r′′(t)|/|~r′|2.19)T FThe cur ve ~r(t) = hcos(t) sin(t), sin(t) sin(t), sin(t)i is located on a sphere.20)T FThe sur fa ce x2+ y2+ z2= 4z − 3 is a sphere of radius 1.Total2Problem 2) ( 10 po ints) No justifications are needed in this problem.a) (2 points) Match the graphs of the functions f(x, y). Enter O, if there is no match.I II IIIFunction f(x, y) = Enter O, I, II or IIIx2||x| − |y||x2+ y2x/(1 + y2)b) (3 points) Match the space curves with their p ar a m et r iz at i on s ~r(t). Enter O, if there is nomatch.I II IIIParametrization ~r(t) = O, I,II,III~r(t) = ht cos(t), t sin(t), ti~r(t) = hcos(t), sin(t), ti~r(t) = hsin(t), cos(t), 0i~r(t) = h∈ (3t), cos(2t), cos(t)ic) (2 points) Match t h e functions g with the level surface g(x, y, z) = 1. Enter O, where nomatch.I II IIIFunction g(x, y, z) = 1 O, I,II,IIIg(x, y, z) = x2− y2− z = 1g(x, y, z) = x − y2= 1g(x, y, z) = x2− y2+ z2= 1g(x, y, z) = x2+ y2+ z2= 1d) (3 poi nts) Match the surface with the parametrization. Enter O, where no match.I II IIIFunction g(x, y, z) = O,I,II,III~r(s, t) = ht, s, tsi~r(s, t) = ht2+ s2, s, ti~r(s, t) = ht, t cos(s) , t sin(s)i~r(s, t) = ht, s, tiProblem 3) ( 10 po ints) No justifications are needed in this problem.a) (2 points) Translate from polar to Cartesian coordinates or back:3Polar coordinates (θ, r) = Cartesian coordinates (x, y) =(π/2, 1)(1, 1)b) (2 poi nts) Translate from spherical to Cartesian coordinates or back:Spherical coordinates (θ, φ, ρ) = Cartesian coordinates (x, y, z) =(π/2, π/2, 1)(1, 1, 1)c) (3 points) Match the curves given in pol ar coor d i n a te s. Enter O, if there is no match.I II IIISurface Enter I,II,II, Or = π/4r = −θθ = π/4r = θd) (3 poi nts) Match the surfaces given in spherical coordinates. Enter O, if there is no match.I II IIISurface Enter I,II,III,Oρ = π/4φ = π/4θ = φθ = π/4Problem 4) ( 10 po ints)a) (2 points) Given two points A = (1, 2, 3) and B = (4, 5, 6). Find the midpoi nt M betweenthese two points.b) (5 points) Find the equation ax + by + cz = d of the plane for which every poi nt has equaldistance to both A and B.c) (3 points) Write down a parametrization ~r(t) = hx(t), y(t), z(t)i of the line containing bothA and B.4Problem 5) ( 10 po ints)In this problem you have to find parametrizations of surfaces. The parametrization sh ou l d havethe form~r(u, v) = hx(u, v), y(u, v), z(u, v)i .a) (2 points) Parametrize the paraboloid z = −x2− y2.b) (2 poi nts) Parametrize the ellipsoid x2+ y2/4 + z2/9 = 1.c) (2 points) Parametrize the plane x = y.d) (2 poi nts) Parametrize the cylinder x2+ z2= 9.e) (2 points) Parametrize the cone x2+ y2= z2.Problem 6) ( 10 po ints)In the soccer world championship of 2010, theEnglish team scored a goal against the Germanteam which the referee did n o t see. Assume the ballfollowed the line x = y = ( z − 1)/3 and that thereferee was at position R = (3, 2, 1).Remark: The incidence was a Wimbleton revanche and confirmeda word of wisdom of former player Gary Lineker who said a coupleof years ago: ”Soccer is a game for 22 people that run around, playthe ball, and one referee who makes a slew of mistakes, and in theend, Germany always wins.”a) (4 points) Find a parametrization of the line L which the b a ll followed.b) (6 poi nts) Find the minimal distance of L from the point R.Problem 7) ( 10 po ints)Compute the fol lowing expression s:a) (2 points) the length of the vector h1, 1, 2i,b) (2 poi nts) the cross product h1, 1, 2i × h2, 2, 1i,c) (2 points) the dot product h1, 1, 4i · h 2, 4, 2i,5d) (2 poi nts) the projection Projh1,1,2i(h1, 2, 3i),e) (2 points) the angle between h1, 1, 0i and h0, 1, 1i.Problem 8) ( 10 po ints)a) (3 points) Find the unit tangent vector~T (t) of …
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