MTH 112 Harvard University Vectors and Dot Products Questions answer the questions that arent crossed out with the black marker on the pages below Section

MTH 112 Harvard University Vectors and Dot Products Questions answer the questions that arent crossed out with the black marker on the pages below Section 8.4 Exercises
Write the vector shown in component form.
1.
Given the vectors shown, sketch ū + v, ū – v , and 2ū.
ū
ū
u
3.
Write each vector below as a combination of the vectors ū and v from question #3.
5.
From the given magnitude and direction in standard position, write the vector in
component form.
7. Magnitude: 6, Direction: 45°
Magnitude: 10, Direction: 120°
9. Magnitude: 8, Direction: 220°
blagnitude: 7, Direction: 305°
(6,5)
Find the magnitude and direction of the vector.
11. (0,4)
• (-3,0)
15. (-2,1)
19. (-4,-6)
•(3,7)
-4)
Using the vectors given, compute ü +ö, ü – , and 2ü – 3v.
21. ü = (2, – 3), v = (1,5)
22. i =-343
27. Three forces act on an object: F = (-8, – 5), F, = (0,1), F, = (4, – 7). Find the net
force acting on the object.
Three forces act on an object: F = (2,5), F, = (8,3), F; = (0, – 7). Find the net force
acting on the object.
29. A person starts walking from home and walks 3 miles at 20° north of west, then 5
miles at 10° west of south, then 4 miles at 15° north of east. If they walked straight
home, how far would they have to walk, and in what direction?
A person starts walking from home and walks 6 miles at 40° north of east, then 2
miles at 15° east of south, then 5 miles at 30° south of west. If they walked straight
home, how far would they have to walk, and in what direction?
31. An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing
from the southwest at 80 km/hr. How many degrees off course will the plane end up
flying, and what is the plane’s speed relative to the ground?
An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing
from the northwest at 50 km/hr. How many degrees off course will the plane end up
flying, and what is the plane’s speed relative to the ground?
33. An airplane needs to head due north, but there is a wind blowing from the southwest
at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due
north, the pilot will need to fly the plane how many degrees west of north?
37. Two children are throwing a ball back and forth straight across the back seat of a car.
The ball is being thrown 10 mph relative to the car, and the car is travelling 25 mph
down the road. If one child doesn’t catch the ball and it flies out the window, in what
direction does the ball fly (ignoring wind resistance)?
Section 8.5 Exercises
Two vectors are described by their magnitude and direction in standard position. Find
the dot product of the vectors.
1. Magnitude: 6, Direction: 45°; Magnitude: 10, Direction: 120°
2. Magnitude: 8, Direction: 220°; Magnitude: 7, Direction: 305°
Find the dot product of each pair of vectors.
3. (0,4); (-3,0)
46,5), (3,7)
5. (-2,1);(-10,13)
2,-5); (8, – 4)
Find the angle between the vectors
7. (0,4); (-3,0)
9. (2,4); (1, -3)
2)
7,10)
13. Find a value for k so that (2,7) and (k,4) will be orthogonal.
Ha tud a value tor (-3,5) and (2,k) will be orthogonal.
15. Find the magnitude of the projection of (8, – 4) onto (1, -3).
+6. Tinų the magnitude of the projection of (2,7) onto (4,5).
17. Find the projection of (-6,10) onto (1, – 3).
end the projection of (0,4) onto (3,7).
19. A scientist needs to determine the angle of reflection when a laser hits a mirror. The
picture shows the vector representing the laser beam, and a vector that is orthogonal
to the mirror. Find the acute angle between these, the angle of reflection.
20. triangle has coordinates at A: (1,4), B: (2,7), and C: (4,2). Find the angle at point B.
21. A boat is trapped behind a log lying parallel to the
dock. It only requires 10 pounds of force to pull
the boat directly towards you, but because of the
log, you’ll have to pull at a 45° angle. How much
force will you have to pull with? (We’re going to
assume that the log is very slimy and doesn’t
contribute any additional resistance.)
23. Find the work done against gravity by pushing a 20 pound cart 10 feet up a ramp that
is 10° above horizontal. Assume there is no friction, so the only force is 20 pounds
downwards due to gravity.
and the work done against gravity by pushing a 30 pound cart 15 feet up a ramp that
is 8° above horizontal. Assume there is no friction, so the only force is 30 pounds
downwards due to gravity.
359
25. An object is pulled to the top of a 40 foot ramp that forms a 10°
angle with the ground. It is pulled by rope exerting a force of
120 pounds at a 35° angle relative to the ground. Find the
work done.
10°

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