7/8/2010 FIRST HOURLY PRACTICE IV Maths 21a, O.Knill, Summer 2010Name:• Start by writing 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, bo oks, calculators, comput ers, or other electronic aids can be allowed.• You have 90 minutes time to complete your work.1 202 103 104 105 106 107 108 109 1010 1011 10Total: 1201Problem 1) (20 points) No justifications are needed.1)T FThe vector ~v = (1, 2, −4) is perpendicular to the plane 4x + 2y + 2z = 100.2)T FWith~i = (1, 0, 0),~j = (0, 1, 0),~k = (0, 0, 1), the formula (~i ×~j) ×(~j ×~i) =~0holds.3)T FThe equations x/5 = y/7 = z/8 describe a line which contains the origin(0, 0, 0).4)T FThe vectors ~u = (3, −2, 1) and~OQ with O = (0, 0, 0) and Q = (−6, 4, 2)are parallel.5)T FIf ~u+~v and ~u−~v are perpendicular, then the vectors ~u and ~v have the samelength.6)T FThe two vectors (2, 3, 0) and (6, −4, 5) are orth ogonal.7)T FFor any two vectors ~v, ~w, one has |~v + ~w|2= |~v|2+ |~w|2.8)T FThe surface x2− y2+ z2= 1 is called a one-sheeted hyperboloid.9)T FThe set of points which have distance 2 from the x-axis is a cylinder.10)T FIf in spherical coordinates a point is given by (ρ, θ, φ) = (2, π/2, π/2), th enits Euclidean coordinates is (x, y, z) = (0, 2, 0).11)T FThe triangle defined by the three points (−1, 0, 2), (−4, 2, 1), (1, −1, 2) hasa right angle.12)T FThe curvature of the curve ~r(t) = hx(t), y(t), z(t)i = (5, 3 sin(t), 3 cos(t)) is1/3.13)T FThe length |~v − ~w| of the difference ~v − ~w of two parallel vectors ~v, ~w isalways equal to the difference |~v| − |~w| of the lengths of the vectors.14)T FThe volume of a parallelepiped spanned by (1, 0, 0), (0, 1, 0) and (1, 1, 1) isequal to 1/3.15)T FThe equation x2= y + z2describes an elliptic parab oloid.16)T FIn spherical coordinates, the equation cos(θ) = sin(θ) is the plane x−y = 0.17)T FIf |~v × ~w| = 0 then ~v =~0 or ~w =~0 or ~v = ~w.18)T FThe vector projection of the vector (1, 1, 1) onto the vector (0, 2, 0) is(0, 1, 0).19)T FThe point given in spherical coordinates as (ρ, θ, φ) = (√8, 3π/2, π/2) is inCartesian coordinates the p oint (x, y, z) = (2, −2, 0).20)T FIf g(x, y, z) = 0 is an implicit equation, then ~r(u, v) =(u, v, g(u, v, g(u, v, 1))) is a parametrization of the surface.Total2Problem 2) (10 points)Match the equations g(x, y, z) = d with the surfaces.I II IIIIV V VIEnter I,II,III,IV,V,VI here Equationx2+ 2z2− y2= 0y2+ z2+ 4x2= 1y2− z2− x2= −1x2− y2− z2= 1y2− z2− x = 1−y2− z2+ x = 1Problem 3) (10 points)3Match the equation with their graphs. No justifications are needed.I IIIII IVV VI4Enter I,II,III,IV,V or VI here Equationz = |x| − |y|z = y2log(x)z = cos(5(x − y))z = x + 2yz = x/(2 + sin(xy))z = x4− y4Problem 4) (10 points)Find the equation of the plane containing A = (1, 1, 0) which is perpendicular to the planesx + y + z = 1andx −y − z = 1 .Problem 5) (10 points)Find the distance between the point P = (1, 0, −1) and the line which contains the pointsA = (1, 1, 1) and B = (0, 2, 1).To do so:a) (4 points) Find first a parametrization ~r(t) = A + t~v of the line.b) (6 points) Now find the distance.Problem 6) (10 points)The angle between two p lanes is defined as the angle between the two normal vectors ofthe planes. Given the planes x − z = 1 and y + z = 3.a) (2 point) Find the normal vector ~v to the first plane and the normal ~w to the secondplane.b) (2 points) Find the angle α between the two planes.5c) (2) points) Verify that P = (4, 0, 3) and Q = (1, 3, 0) are on the intersection of the twoplanes.d) (2 points) Find a parametrization of the line of intersection of the two planes.e) (2 points) Find the symmetric equation of the line of intersection of the two planes.Problem 7) (10 points)Find the implicit equationax + by + cz = dof the plane which contains the line~r(t) = P + t~v = (−1, 1, 1) + t(3, 4, 5)and the point Q = (7, 7, 9).Problem 8) (10 points)a) (6 points) Find the surface area of the parallelepiped which contains the points A =(0, 0, 0), B = (1, 1, 0), C = (0, 1, 1), D = (1, 0, 1).b) (4 points) Find the volume of the solid.ABCDProblem 9) (10 points)6a) (4 points) We define a scalar valued function which has as argument two vectors:f(~v, ~w) =|~v × ~w|~v · ~wWhat is f((1, 0, 0), (1, 1, 0))?b) (6 points) The function f(~v, ~w) turns out t o be a function of the angle α between ~v and~w only. What is this function?Problem 10) (10 points)The set of points P for which the distance from P to A = (1, 2, 3) is equal to the distancefrom P to B = (5, 8, 5) forms a plane S.a) Find the equation ax + by + cz = d of the plane S.b) Find the distance from A to S.Problem 11) (10 points)Find the distance between the z-axis and the lineL :x + 24=y −
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