7/8/2010 FIRST HOURLY PRACTICE III Maths 21a, O.Knill, Summer 2010Name:• Start by printing your name in the above box.• Try to answer each question on the same page as the question is asked. If needed, 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 90 minutes time to complete your work (the actual exam will have only 10questions)1 202 103 104 105 106 107 108 109 1010 1011 10Total: 1201Problem 1) (20 points)Circle for each of the 20 questions the correct letter. No justifications are needed.1)T FThe set of points in the plane which satisfy x2−y2= −10 is a curve calledhyperbola.2)T FThe length of the sum of two vectors in space is always the sum of thelength of the vectors.3)T FFor any three vectors ~u, ~v, ~w, the identity ~u · (~v × ~w) = ( ~w ×~v) · ~u holds.4)T FThe set of points which satisfy x2+ 2x + y2− z2= 0 is a cone.5)T FIf P, Q, R are 3 different points in space that don’t lie in a line, then~P Q×~RQis a vector orthogonal to the plane containing P, Q, R.6)T FThe line ~r(t) = (1 + 2t, 1 + 3t, 1 + 4t) hits the plane 2x + 3y + 4z = 9 at aright angle.7)T FA surface which is given as r = sin(z) in cylindrical coordinates stays thesame when we rotate it around the y axis.8)T FFor any two vectors, ~v × ~w = ~w ×~v.9)T FIf |~v × ~w| = 0 for all vectors ~w, then ~v =~0.10)T FIf ~u and ~v are orthogonal vectors, then (~u ×~v) ×~u is parallel to ~v.11)T FEvery vector contained in the line ~r(t) = (1 + 2t, 1 + 3t, 1 + 4t) is parallelto the vector (1, 1, 1).12)T FIf in spherical coordinates a point is given by (ρ, θ, φ) = (2, π/2, π/2), thenits rectangular coordinates are (x, y, z) = (0, 2, 0).13)T FThe set of points which satisfy x2− 2y2− 3z2= 0 form an ellipsoid.14)T FIf ~v × ~w = (0, 0, 0), then ~v = ~w.15)T FThe set of points in R3which have d istance 1 from a line form a cylinder.16)T FIf in rectangular coordinates, a point is given by (1, 0, 1), then its sphericalcoordinates are (ρ, θ, φ) = (√2, π/2, −π/2).17)T FIn spherical coordinates, the equation cos(θ) = sin(θ) defines the planex − y = 0.18)T FFor any three vectors ~a,~band ~c, we always have (~a ×~b) ·~c = −(~a ×~c) ·~b.19)T FIf |~v × ~w| = 0 then ~v = 0 or ~w = 0.20)T FTwo nonzero vectors are parallel if and only if their cross product is~0.2Problem 2a) (5 points)Match the surfaces with their parameterization ~r(u, v) or the implicit description g(x, y, z) = 0.Note that one of the surfaces is not represented by a formula. No justifications are needed inthis problem.I II II IIV V VIEnter I,II,III,IV,V,VI here Equation or Parameterization~r(u, v) = ((2 + sin(u)) cos(v), (2 + sin(u)) sin(v), cos(u))~r(u, v) = (v, v − u, u + v)~r(u, v) = (u2, vu, v)x2− y2+ z2− 1 = 0~r(u, v) = (cos(u) sin(v), cos(v), sin(u) sin(v))Problem 2b) (5 points)Match the contour maps with the corresponding functions f(x, y) of two variables. No justifi-cations are needed.3I II II IIV V VIEnter I,II,III,IV,V or VI here Function f(x, y)f(x, y) = sin(x)f(x, y) = x2+ 2y2f(x, y) = |x| + |y|f(x, y) = sin(x) cos(y)f(x, y) = xe−x2−y2f(x, y) = x2/(x2+ y2)Problem 3) (10 points)Match the equation with their graphs and justify briefly your choice:424681024681000.511.522468-4-2024-4-202400.51-4-2024I II-2-1012-2-1012-1-0.500.51-2-101-1-0.500.51-1-0.500.51012345-1-0.500.5II I IVEnter I,II,III,IV here Equation Short Justificationz = sin(3x) cos(5y)z = cos(y2)z = log(x)z = x/(x2+ y2)Problem 4) Distances (10 points)Let L be the linex = 1 + 2t, y = −3t, z = tand let S be the plane x + y + z = 2.a) Verify that L and S have no intersections.b) Compute the distance between the line L and plane S.Hint. Just take any point P on the line and compute the distance from the line to the plane.5Problem 5) (10 points)a) (5 points) Find the area of the parallelogram P QSR with cornersP = (0, 0, 0), Q = (1, 1, 1), R = (1, 1, 0), S = (2, 2, 1) .b) (5 points) Find the volume of the pyramid which has as the base the parallelogram P QRSand has a fifth vertex at T = (3, 4, 3).Problem 6) (10 points)Find the distance between the two lines~r1(t) = ht, 2t, −tiand~r2(t) = h1 + t, t, ti .Problem 7) (10 points)Given the vectors v = (1, 1, 0) and w = (0, 0, 1) and the point P = (2, 4, −2). Let Σ be theplane which goes through the origin and contains the vectors ~v and ~w.a) Determine the distance from P to the origin.b) Determine the distance from P to the plane Σ.Problem 8) (10 points)6In this problem, it is enough to describe the surface with words.a) (3) Identify the surface whose equation is given in spherical coordinates as φ = π/6.b) (3) Identify the surface whose equation is given in spherical coordinates as θ = π/2.c) (2) Identify the surface, whose equation is given in cylindrical coordinates by z2= r.d) (2) Identify the surface, whose equation is given in cylindrical coordinates as r cos(θ) = 1Problem 9) (10 points)a) (3 points) Let S be the surface g(x, y, z) = x2− z − y2= 1. Find a parametrization~r(u, v) = (x(u, v), y(u, v), z(u, v))of this surface.b) (3 points) Write down the parametrization~r(u, v) = (x(u, v), y(u, v), z(u, v))of the part of the unit sphere x2+ y2+ z2= 1 which satisfies z ≥√3/2 and also indicate thedomain R of the parametrization.c) (4 points) Let S be the surface given in cylindrical coordinates as r = 2 + sin(z). Find aparameterization~r(u, v) = (x(u, v), y(u, v), z(u, v))of the surface.Problem 10) (10 points)Remember that a parameterization of a surface describes the points (x, y, z) of the surfacein the form ~r(u, v) = (x, y, z) = (x(u, v), y(u, v), z(u, v)). What surfaces do the followingparameterizations represent? Find in each case an implicit equation of the form g(x, y, z) = cwhich is equivalent.a) (3) ~r(u, v) = (cos(u), sin(u), v)b) (3) ~r(u, v) = (u + v, v − u, u + 2v)7c) (2) ~r(u, v) = (v cos(u), v sin(u), v)d) (2) ~r(u, v) = (cos(u) sin(v), sin(u) sin(v), cos(v)).Problem 11) (10 points)Find an equation for the plane that passes through the origin and whose normal vector isparallel to the line of intersection of the planes 2x + y + z = 4 and x + 3y + z =
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