Math 227-104 (CRN 24680)
Carter
Test 1 Spring 2015
General Instructions. Write your name on only the outside of your blue books. Do not
write on this test sheet, do all of your work inside your blue books. Write neat complete
solutions to each of the problems in the blue book. Please put your test sheet inside the blue
book as you leave. There are 110 points.
For a sandwich, try avocado, tomato, cheese, and mayo on a whole grain bread.
1. (10 points) Compute the dot product of the two vectors:
√
√
~v = 2ı̂ − 4̂ + 5k̂, ~u = −2ı̂ + 4̂ − 5k̂.
2. (20 points) Compute the volume of the parallelepiped that has ~0 and the vectors ~u, ~v ,
and w
~ as corners.
~u = ı̂ + ̂ + k̂,
~v = ı̂ + 2̂ − k̂,
w
~ = 7̂ − 4k̂.
What are the other four vertices of this figure?
3. (15 points) Find a vector that is perpendicular to the plane that contains the points
P~ = (2, −2, 1),
~ = (3, −1, 2),
Q
~ = (3, −1, 1).
R
4. (10 points) Give the equation of the plane that contains the point P~ = (2, 1, 3) and is
perpendicular to the vector ~n = ı̂ + 2̂ − 4k̂.
5. (15 points) Draw the intersection of the quadratic surface
x2 + y 2 − z 2 = 1
with each of the coordinate planes x = 0, y = 0, and z = 0. Either draw or give a
concise description (name it) of this quadratic surface.
1
6. Consider the helix
r(t) = cos (t)ı̂ + sin (t)̂ + tk̂.
(a) (5 points) Determine the arc length
Z
2π
||~v (t)|| dt
s(2π) =
0
(b) (5 points) Determine the unit tangent vector
T~ = ~v /||~v ||.
(c) (5 points) Determine the unit normal vector
~
~ = dT /dt .
N
||dT~ /dt||
~ = T~ × N
~ when t = π/2.
(d) (5 points) Determine the binormal B
7. Let ~r(t) denote the position vector of a particle in space (you won’t need to use ~r(t)
explicitly), and let ~v (t) denote its velocity (~v (t) = ~r 0 (t)). The unit tangent vector T~
is defined above.
(a) (5 points) Show that
dT~
1
=
dt
||~v ||3
(~v · ~v )~a − (~v · ~a)~v
(b) (5 points) Use the identity
~x × (~y × ~z) = (~x · ~z)~y − (~x · ~y )~z,
to show that
1
dT~
=
~v × (~a × ~v ).
dt
||~v ||3
(c) (5 points) Compute
||~v × (~a × ~v )||.
Hint: ~v ⊥ (~a × ~v ).
(d) (5 points) Conclude that
1 dT~ (~a × ~v ) .
=
||~v || dt ||~v ||3 2
.