MATH 21A Harvard 21a Fall 14Midterm 1Exam
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9/30/2014 FIRST HOURLY Math 21a, Fall 2014
MWF 9 Oliver Knill
MWF 9 Chao Li
MWF 10 Gijs Heuts
MWF 10 Yu-Wen Hsu
MWF 10 Yong-Suk Moon
MWF 11 Rosalie Belanger-Rioux
MWF 11 Gijs Heuts
MWF 11 Siu-Cheong Lau
MWF 12 Erick Knight
MWF 12 Kate Penner
TTH 10 Peter Smillie
TTH 10 Jeﬀ Kuan
TTH 10 Yi Xie
TTH 11:30 Jeﬀ Kuan
TTH 11:30 Jameel Al-Aidroos
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Problem 1) (20 points) No justiﬁcations are needed.
1) T F The surface −x2+y2+z2=−1 is a one-sheeted hyperboloid.
2) T F The equation y= 3x+ 2 in space deﬁnes a plane.
3) T F Whenever |~r ′(t)|= 1 then |~
4) T F The length of the vector projection Proj~v(~w) is smaller than or equal to the
length of ~w.
5) T F The velocity vector of ~r(t) = ht, t, tiat time t= 2 is the same as the velocity
vector at t= 1.
6) T F If ~v ×~w =~w ×~v then ~v, ~w are parallel.
7) T F The vector h−2,1,0iis perpendicular to the line h1 + t, 2t, 3ti.
8) T F The point given in spherical coordinates as ρ= 3, φ = 0, θ =πis the same
point as the point ρ= 3, φ = 0, θ = 0.
9) T F The parametrized curve ~r(t) = h0,3 cos(t),5 sin(t)iis an ellipse.
10) T F The curvature of the line ~r(t) = ht, t, tiis √3 everywhere.
11) T F If |~v ×~w|=~v ·~w then either ~v is parallel to ~w or perpendicular to ~w.
12) T F If the dot product between two unit vectors ~v, ~w is −1, then ~v =−~w.
13) T F Writing ~
k=h0,0,1i, we have |(~
k×~v)×~w| ≤ |~v||~w|for all vectors ~v, ~w.
14) T F The curvature of a curve ~r(t) is given by κ(t) = |~
T′(t)|/|~r ′(t)|. If |~
for all times, then κ(t) = |~r ′′(t)|.
15) T F The arc length of the curve hsin(t/2),0,cos(t/2)ifrom t= 0 to t= 2πis
equal to 2π.
16) T F If L, K are skew lines in space, there is a unique plane which is equidistant
from L, K.
17) T F The curve ~r(t) = ht, t2,1−tiis the intersection curve of a plane x+z= 1
18) T F The lines ~r1(t) = h5 + t, 3−t, 2−tiand ~r2(t) = h6−t, 2 + t, 1−2tiintersect
at (6,2,1) perpendicularly.
19) T F The vector h3/13,12/13,4/13iis a unit vector.
20) T F ~v ×(~v ×~u) = ~
0 for all vectors ~u, ~v.
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