10/18/2005, FIRST HOURLY Math 21a, Fall 2005Name:MWF9 Ivan PetrakievMWF10 Oliver KnillMWF10 Thomas LamMWF10 Micha el ScheinMWF10 Teru Yo shidaMWF11 Andrew DittmerMWF11 Chen-Yu ChiMWF12 Kathy PaurTTh10 Valentino TosattiTTh11.5 Kai-Wen LanTTh11.5 Jeng-Daw Yu• Please mark the box to the left which lists your section.• Do not detach pages from this exam packet or unstaplethe packet.• Please write neatly. Answers which the grader can notread will not receive credit.• No notes, books, calculators, computers, or other elec-tronic aids can be used.• You have 90 minutes time to complete your work.1 202 103 104 105 106 107 108 109 10Total: 100Problem 1) TF questions (20 points)Mark for each of the 20 questions the correct letter. No justifications are needed.1)T FThe vectors h3, −2, 1i and h−6, 4, −2i are parallel.2)T FIf |~v × ~w| = 0 then ~v =~0 or ~w =~0.3)T FThe surface x2+ 4y2= z2+ 1 has two sheets (that is, it consists of twosurfaces which are not connected to each other).4)T FThe surface 4x2− 4x + y2− 2y − 120 = −z2is an ellipsoid.5)T FThe parametrized lines ~u(t) = h1+2t, −2−5t, 1+ti and ~v(t) = h4−4t, −12+10t, 3 − 2ti are the same line.6)T FThe surface sin(x) = z contains lines which are parallel to the y-axis.7)T FIf ~u · ~v = 0, ~v · ~w = 0 and ~v is not the zero vector, then ~u · ~w = 0.8)T FThe curvature of a curve depends upon the speed at which one travels uponit.9)T FTwo lines in space that do not intersect must be parallel.10)T FThe intersection of the ellipsoid x2/3+y2/4+z2/3 = 1 with the plane y = 1is a circle.11)T FThe line ~r(t) = h1 + 2t, 1 + 2t, 1 − 4ti hits the plane x + y + z = 9 at a rightangle.12)T FA line in space can intersect an elliptic paraboloid in 4 points.13)T FThere is a quadric which is a hyperbola when int ersected with the planez = 0, which is a hyperbola when intersected with the plane y = 0 andwhich is a parabola when intersected with x = 0.14)T FThe vector ~u × (~v × ~w) is always in the same plane as ~v and ~w.15)T FIf ~u × ~v = 0 and ~u · ~v = 0, then one of the vectors ~u and ~v is zero.16)T FThe triple scalar product ~u·(~v× ~w) of three vectors ~u, ~v, ~w can be a negativenumber.17)T FIf the velocity vector ~r′(t) and the acceleration vector ~r′′(t) of a curve areparallel at time t = 1, then the curvature κ(t) of the curve is zero at timet = 1.18)T FIf the speed of a parametrized curve is constant over time, then the curvatureof the curve ~r(t) is zero.19)T FThe scalar projection of a vector ~v onto a vector ~w is always equal to thescalar projection of ~w onto ~v.20)T FThe velocity vector of a parametric curve ~r(t) always has length 1.Problem 2) (10 points)Match t he equation with their graphs. No justifications are needed.I IIIII IVEnter I,II,III,IV here Equationz = sin(5x) cos(2y)z = cos(y2)z = e−x2−y2z = exProblem 3) (10 points)Match t he equations with the surfaces.I II IIIIV V VIEnter I,II,III,IV,V,VI here Equationx2− y2− z2= 1x2+ 2y2= z22x2+ y2+ 2z2= 1x2− y2= 5x2− y2− z = 1x2+ y2− z = 1Problem 4) (10 points)a) (7 points) Find a parametric equation for the line which is the intersection of the two planes2x − y + 3z = 9 and x + 2y + 3z = −7.b) (3 points) Find a plane perpendicular to bo th planes and which passes through the pointP = (1, 1, 1).Problem 5) (10 points)Given the vectors ~v = h1, 1 , 0i and ~w = h0, 0, 1i a nd the point P = (2 , 4, −2). Let Σ be theplane which goes through the origin which contains the vectors ~v and ~w. Let S be the unitsphere x2+ y2+ z2= 1.a) (6 points) Compute the distance f r om P to the plane Σ.b) (4 points) Find the shortest distance from P to the sphere S.Hint for b): Find first the distance from P to the origin O = (0, 0, 0).Problem 6) (10 points)a) (6 points) Find an equation for the plane through the points A = ( 0 , 1, 0), B = ( 1 , 2, 1) andC = (2, 4, 5).b) (4 points) Given a n additional point P = (−1, 2, 3), what is the volume of the tetrahedronwhich has A, B, C, P among its vertices.A useful fact which you can use without justification in b): the volume of the tetrahe-dron is 1/6 of the volume of the parallelepiped which has AB, AC, and AP among its edges.Problem 7) (10 points)The parametrized curve ~u(t) =< t, t2, t3> (known as the ”twisted cubic”) intersects theparametrized line ~v(s) = h1 + 3s, 1 − s, 1 + 2si at a point P . Find the angle of intersection.Problem 8) (10 points)Let ~r(t) be the space curve ~r(t) = (log(t), 2t, t2), where log(t) is the na tura l logarithm (denotedby ln(t) in some textbooks).a) What is the velocity and what is the acceleration at time t = 1?b) Find the length of the curve from t = 1 to t = 2.Hint: you should end up with a final integral which does not involve any square roots andwhich you can solve.Problem 9) (10 points)A planar mirror in space contains the point P = (4, 1, 5) and is perpendicular to the vector~n = h1, 2, −3i. The light ray~QP = ~v = h −3, 1, −2i with source Q = (7, 0, 7) hits the mirrorplane at the point P .a) (4 points) Compute the projection ~u =~Proj~n(~v) of ~v onto ~n.b) (6 points) Identify ~u in the figure and use it to find a vector parallel to the reflected
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