MI IV
April 14, 2011
Vectors - Getting to the Point
An Opportunity To Demonstrate Knowledge
Name _________________
Mods _________
Calculators are probably useful on some of these problems. So feel free to use one as you
wish. However, remember, you are trying to show the depth and breadth of your knowledge of
vectors! Present appropriate justification and labels for all of your answers.
1. Given a  3,3 and b  1, 3 .
2 pts.
4 pts.
a.
Sketch and label a , b , and projb a .
3 pts.
b.
Compute projb a
2. A vector is drawn from the origin so that it makes an angle theta with the positive x-axis, and
the terminal point (head) of the vector is on the unit circle. Find the components of this
vector.
<
,
>
3. In the drawing to the right, u  AB and v  BD . C is the midpoint of BD and
2
AF  AD . State the following in terms of u and v .
3
2 pts.
a. AD 
2 pts.
b. AC 
B
C
A
F
3 pts.
c. CF 
1
D
MI IV
April 14, 2011
4 pts.
3 pts.
3 pts.
Name _________________
Mods _________
4. Given u = <3, –11>, v = <–3, 1> and w = <5, 2>, find values of a and b so that u = a v + b w.
5. If v = 6, 3 and w = 1, 2 , find 2( v )  w .
6. Find the components of the transformed vector when vector <2, -3> is multiplied by the
 0.34 0.94 
transformation matrix 
 . Describe the transformation of the vector and set up
 0.94 0.34 
the matrix equation to explain your answer.

2 pts.
Vectors - Getting to the Point
An Opportunity To Demonstrate Knowledge
 
7. Let u  12i  5 j . Find a vector that has the same direction as u but has length of 3.
2
MI IV
April 14, 2011
4 pts.
Name _________________
Mods _________
8a. Given v = x  8, 3 and w= 1  x , -x , find the value(s) of x so that w is orthogonal to v .
2 pts.
2 pts.
Vectors - Getting to the Point
An Opportunity To Demonstrate Knowledge
9a.
b. What can you say about the angle between v and w if the vectors are collinear? Explain.
Sketch v = 3, –2, 4 .
z
y
x
3 pts.
b.
Find the angle between v = 3, –2, 4 and w = 1, 4, –3 to the nearest tenth of a degree.
3
MI IV
April 14, 2011
Vectors - Getting to the Point
An Opportunity To Demonstrate Knowledge
Name _________________
Mods _________
10. State whether the following operations result in a scalar or a vector:
1 pt.
a b
1 pt.
v  2u ____________________________
1 pt.
proja b ____________________________
5 pts.
_____________________________
11. A plane is heading N70°W at 340 mph. It encounters a wind of 36 mph heading S80°E.
Determine the direction and resulting speed of the plane and explain your answer with a well
labeled diagram.
12. State whether the following statements are True or False:
(1 pt. each)
_____ If a and b are unit vectors, the the measure of the angle between a and b is
cos 1 a  b .


_____ A boat is traveling at 12 knots S30°W. The component form of the vector describing the
boat is 12sin  120  ,12 cos  120  .
_____ Let k be a real number. For any vectors v and u , k  v  u   kv  ku .
4
MI IV
April 14, 2011
Vectors - Getting to the Point
An Opportunity To Demonstrate Knowledge
Name _________________
Mods _________
13. A river is 86 meters wide and has a current of 20 meters/minute. A river boat pilot wants to
head straight across the river. The boat can travel 100 meters/minute in still water.
a)
Determine the angle to the bank the pilot should head the boat upstream to arrive directly
3 pts.
across the river. Give your direction in terms of bearing.
2 pts.
b) Determine the time it takes her to arrive on the opposite bank of the river.
5