Math 2210-1
Homework 4
Due Wednesday June 9
Show all work. Please box
√ your answers. Be sure to write in complete sentences when appropriate. Also,
I prefer exact answers like 2 instead of 1.414. Note that a symbol indicates that graph paper might be
useful for that problem.
Lines and Curves in Three-Space
1. Give the parametric eqution of the line through the points (2,-1,-5), (7,-2,3).
2. Write the parametric equations and the symmetric equations for the line through the point (-2,2,-2)
and parallel to the vector < 7, −6, 3 >.
3. Give the symmetric equations of the line of intersection of the planes x − 3y + z = −1, 6x − 5y + 4z = 9.
x−1
y−2
z−4
x−2
y−1
z+2
=
=
and
=
=
intersect, and find the
−4
3
−2
−1
1
6
equation of the plane that they determine.
4. Show that the lines
5. Find the distance between the skew lines x = 1 + 2t, y = −3 + 4t, z = −1 − t and x = 4 − 2t, y =
1 + 3t, z = 2t. (Problem 21 in section 14.4 gives a method to do this, but try to figure it out on your
own first.)
6. Find the equation of the plane perpendicular to the curve r(t) = t sin ti + 3tj + 2t cos tk at t =
π
2.
Velocity, Acceleration, and Curvature
7. Find the velocity, acceleration, and speed of the particle with position vector given by ~r(t) = t6~i +
(6t2 − 5)6~j + t~k at t = 1.
8. Show that if the speed of a moving particle is constant its acceleration vector is always perpendicular
to its velocity vector.
9. Find the curviture, the unit tangent vector, the principal normal, and the binormal of ~r(t) = e7t cos 2t~i+
e7t sin 2t~j + e7t~k at t = π3 .
10. Find the tangential and normal vector components, aT and aN , of the acceleration vector at any time
t of ~r(t) = t sin t~i + t cos t~j + t2~k.
~ and B
~ for ~r(t) = t~i + 1 t3~j + t−1~k at t = 1.
11. Find the vectors T~ , N
3
12. Consider the motion of a particle along a helix given by ~r(t) = sin t~i + cos t~j + (t2 − 3t + 2)~k, where
the ~k component measures the height in meters above the ground at t ≥ 0.
(a) Does the particle ever move downward?
(b) Does the particle ever stop moving?
(c) At what times does it reach a position 12 meters above the ground?
Surfaces in Three-Space
13. The surface in the figure below is the graph of the function z = f (x, y) for positive x and y.
(a) Suppose y is fixed and positive. Does z increase or decrease as x increases? Sketch a graph of z
against x.
(b) Suppose x is fixed and positive. Does z increase or decrease as y increases? Sketch a graph of z
against y.
14. Match the following functions with their graphs below.
(a) z =
1
x2 + y 2
(b) z = −e−x
2
−y 2
(c) z = x + 2y + 3
(d) z = −y 2
(e) z = x3 − sin y
(A)
(C)
(B)
(E)
(D)
15. You like pizza and you like coke. Which of the graphs below represents your happiness as a function
of how many pizzas and how much coke you have if
(a) There is no such thing as too many pizzas and too much coke?
(b) There is such a thing as too many pizzas or too much coke?
(c) There is such a thing as too many pizzas, but no such thing as too much coke?
(A)
(C)
(B)
(D)
16. Imagine a single wave traveling along a canal. Suppose x is the distance from the middle of the canal,
t is the time, and z is the height of the water above equilibrium level. The graph of z as a function of
x and t is shown below.
(a) Draw the profile of the wave for t = −1, 0, 1, 2. (Show the x-axis to the right and the z-axis
vertically.)
(b) Is the wave traveling in the direction of increasing or decreasing x?
(c) Sketch a surface representing a wave traveling in the opposite direction.
17. Name and sketch the graph of each of the following equations in three-space.
(a) y 2 + z 2 = 15
(b) x2 = 3y
(c) 2x2 − 16z 2 = 0
(d) 9x2 − y 2 + 9z 2 − 9 = 0
(e) 6x − 3y = π