10/16/2002 FIRST HOURLY (PROTO 4) Math 21aName:MWF9 Sasha BravermanMWF10 Ken ChungMWF10 Jake RasmussenMWF10 WeiYang QuiMWF10 Spiro KarigiannisMWF11 Vivek MohtaMWF11 Jake RasmussenMWF12 Ken ChungTTH10 Oliver KnillTTH11 Daniel Goroff• Start by printing your name in the above box and checkyour section in the box to the left.• Try to answer each question on the same page as thequestion is asked. If needed, use the back or next emptypage for work. If you need additional paper, write yourname on it.• Do not detach pages from this exam packet or unstaplethe packet.• Please write neatly. Answers which are illegible for thegrader can not be given credit.• No notes, books, calculators, computers or other elec-tronic aids are allowed.• You have 90 minutes time to complete your work.1 802 303 304 405 306 307 308 409 40Total: 350Problem 1) (80 points)Circle for each of the 20 questions the correct letter. No justifications are needed.T FThe vector connecting the point (1, 4, 2) with the point (1, 1, 1) isparallel to the vector h0, −6, −2i.T FThe length of the sum of two vectors is the sum of the length ofthe vectors.T FFor any three vectors, ~v · (~w + ~u) = ~w ·~v + ~u ·~v.T FFor any three vectors, (~v × ~w) · ~u = ( ~w ×~v) ·~u.T FFor any three vectors |(~u ×~v) · ~w| = |(~u × ~w) ·~v|.T FThe vectors~i +~j and~k are orthogonal.T FFor any vector ~v one has ~v × (2~v) = 0.T FIf we attach the vector h2, 1, 1i to the point P = (2, 3, 4), the tip ofthe vector points to the point Q = (3, 4, 5).T FThe set of points which have distance 1 from a plane form a singleplane.T F|~v × ~w| = 0 implies ~v = 0 or ~w = 0.T FThe set of points which satisfy x2+ 2x + y2− z2= 0 is a cone.T FIf ~u + ~v and ~u − ~v are orthogonal, then the vectors ~u and ~v havethe same length.T FIf P, Q, R are 3 different points in space that don’t lie in a line, then~P Q ×~RQ is a vector orthogonal to the plane containing P, Q, R.T FThe line ~r(t) = (1 +2t, 1+3t, 1+4t) hits the plane 2x+3y +4z = 9at a right angle.T FIf in rectangular coordinates, a point is given by (1, −1, 0), then itsspherical coordinates are (ρ, θ, φ) = (√2, −π/2, π/2).T FIf the velocity vector of the curve ~r(t) is never zero and alwaysparallel to a constant vector ~v for all times t, then the curve is astraight line.T FThe equation r = 3z in cylindrical coordinates defines a cone.T FThe set of points in the x − y plane which satisfy x2− 2y2= 0 isan ellipse.T FA surface which is given as r = sin(z) in cylindrical coordinatesstays the same when we rotating it around the y axes.T FThe identity |~v · ~w|2+ |~v × ~w|2= |~v|2|~w|2holds for all vectors ~v, ~w.x 4 =2Problem 2) (30 points)Match the equation with their graphs. To do so, it can help to look at the intersection of eachsurface with the xy-plane.I II-4-2024-4-2024-5-2.502.55-4-2024-4-2024-4-2024-5-2.502.55-4-2024III IV-2-1012-2-1012-505-2-1012-4-2024-4-20240.60.70.80.91-4-2024V VI-4-2024-4-2024051015-4-2024-202-202051015-202I,II,III,IV,V or VI? Equationz = sin(x)yz = cos(π(1+x2+y2))2x + 3y + 4z = 0I,II,III,IV,V or VI? Equationz = y3z = x2z = x2+ y23Problem 3) (30 points)Match the equation with their graphs and describe the x-y trace (the intersection of the surfacewith the x − y plane z = 0) with maximally three words in each case.I II IIIIV V VIEnter I,II,III,IV,V,VI here Equation Describe the x-y trace in wordsx2− y2− z2= 1x2+ 2y2= z22x2+ y2+ 2z2= 1x2− y2= 5x2− y2− z = 1x2+ y2− z = 14Problem 4) (40 points)Find the distance between the point P = (1, 0, −1) and the plane which contains the pointsA = (1, 1, 1) and B = (0, 2, 1) and C = (1, 2, 2).To do so:a) Find an equation of the plane.b) Find the distance.Problem 5) (30 points)An ant has gotten into the math department ’surfacecabinet’ and is walking around on one of the models.Her position in cylindrical coordinates isr(t) = 2√tθ(t) = tz(t) = 2tfor 0 ≤ t ≤ 6π.a) What are the parametric equations describing the ant’s path in rectangular coordinates?b) Write an equation (in either rectangular or cylindrical coordinates) which might describethe surface the ant is walking on. Sketch the surface given by your equation, and indicate theant’s path.(Hint: there are many possible surfaces. You may find some easier to draw than others.)Problem 6) (30 points)In the xy-plane, the equation x2− y2= 1 defines a hyperbola. If we rotate this hyperbolaaround the y-axis, we obtain a surface in three dimensional space. Sketch this surface and5write an equation which defines it.Problem 7) (30 points)Let ~a and~b be two vectors in R3. Assume that the length of ~a ×~b is equal to 10. What is thelength of (~a +~b) × (~a −~b)?Problem 8) (40 points)Consider the parametrized curve ~r(t) = (etcos(t), etsin(t), et).a) Find a parametric equation for the tangent line to this curve at t = π.b) Find a scalar equation for the plane perpendicular to the curve at the same point.c) Find the arclength of the segment of the curve for which 0 ≤ t ≤ 1.Problem 9) (40 points)Let C be the curve of the intersection of the elliptical cylinderx225+y29= 1 in three dimensionswith the plane 3z = 4y.a) Find a parametric equation ~r(t) = (x(t), y(t), z(t)) of C.b) Find the arc length of
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