Math 2210-1
Homework 2
Due Wednesday May 26
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.
Vectors in the Plane: Geometric Approach
1. Say whether the given quantity is a vector or a scalar.
(a) The distance from Seattle to St. Louis.
(b) The population of the U.S.A.
(c) The magnetic field at a point on the earth’s surface.
(d) The temperature at a point on the earth’s surface.
2.
Draw two vectors v and w in the plane at some angle to one another and with different magnitudes.
On this drawing, draw the following vectors:
(a) v + w
(b) v − w
(c) 2v
(d) 2v + w
(e) v − 2w
3.
Mark pushes on a post in the direstion S30◦E (30◦ east of south) with a force of 60 pounds.
4.
A 300-newton weight rests on a smooth (friction negligible) inclined plane that makes an angle of
Dan
pushes on the same post in the direction S60◦ W with a force of 80 pounds. What are the magnitude
and direction of the resultant force?
30◦ with the horizontal. What force parallel to the plane will just keep the weight from sliding down
the plane? (Hint: Consider the downward force of 300 newtons to be the sum of two forces, one parallel
to the plane and one perpendicular to it.)
Vectors in the Plane: Algebraic Approach
5. Let a =< 3, −1 >, b =< 1, −1 >, and c =< 0, 5 >. Find each of the following:
(a) −4a + 3b
(b) a · b
(c) (a + b) · c
(d) 2c · (3a + 4b)
(e) |b|b · a
2
(f) |c| − c · c
6. Find the angle between a and b and make a sketch.
(a) a = 4i + 3j, b = −8i − 6j
√
√
(b) a = 3i + j, b = 3i + 3j
7. Find a vector that points in the same direction as i − 3j but has length 2.
8. Find a vector that points in the opposite direction as −3i + 7j but has length 5.
9. Find a vector orthogonal to 2i + 7j.
10. Find the work done by the force F = 3i + 10j newtons in moving an object 10 meters north (i.e. in
the j direction).
11. Find projv u for the following:
(a) u =< 0, 5 >, v =< 3, 4 >
(b) u =< −3, 2 >, v =< 3, 4 >
Vector-Valued Functions and Curvilinear Motion
12.
Below is a map of a race track.
A car drives clockwise around the course, slowing down at the
curves and speeding up along the straight portions of the track. Sketch velocity vectors at the points
P, Q, and R. A second car drives clockwise around the track at constant speed. At what point on the
track does the car have the longest acceleration vector, and in roughly what direction is it pointing?
Remember that acceleration is the rate of change of velocity.
0011 R
P 0011
0011
Q
sin t cos t
7t3
13. Find the limit of lim
i− t j .
t→0
t
e
14. Find the first and second derivative of the following functions:
2
(a) r(t) = (et + e−t )i + 2t j
(b) F(t) where F(t) = f (g(t)), and f (u) = cos ui + e3u j and g(t) = 3t2 − 4.
15. The position of a particle is given by r(t) = a sin ti + 2a cos tj where a > 0. Find the velocity and
acceleration vectors and the speed at the time t = π4 .
Curvature and Acceleration
16. Find the unit tangent vector T(t) and the curvature κ(t) at the point where t = t1 .
(a) u(t) = 4t2 i + 4tj; t1 = 12
(b) r(t) = e−3t i + et j; t1 = 0
17. Sketch the following curves. For the given point, find the curvature and the radius of curvature. Also,
draw the circle of curvature at the point.
(a) y 2 − 4x2 = 20; (2, 6)
2
(b) y = e−x ; (1, 1e )
18. Find the point of the curve y = ln cos x for
−π
2
<x<
π
2
where the curvature is a maximum.
19. A car traveling at constant speed v rounds a level curve, which we take to be a circle of radius R. If
the car is to avoid sliding outward, the horizontal frictional force F exerted by the road on the tires
must at least balance the centrifugal force pulling outward. The force F satisfies F = µmg, where
µ is the coefficient of friction, m is the mass of the car, and g is the acceleration of gravity. Thus
√
mv 2
. Show that vR , the speed beyond which skidding will occur, satisfies vR = µgR and
µmg ≥
R
use this to determine vR for a curve with R = 400 feet and µ = 0.4. Use g = 32 feet per second per
second.