Pre-AP/GT Pre-Calculus Assignment Sheet
Unit 10 - Vectors
February 28th – March 9th, 2017
Date
Tuesday
2/28
Wednesday
3/1
Objective
Basic Vector Operations and Represent Graphically
Assignment
Finish Pages 2-5
Basic Vectors (Day2)
Reteach Unit 9 – Law of Sines, Cosines
Finish Pages 6-8
Basic Vectors (Day 3)
Pages 11-12
Retention Quiz Unit 9 – Law of Sines, Cosines
Finish Pages 13-15
Monday
3/6
Dot Product of Vectors, Sum of Component Vectors
Page 17- top of 18
Tuesday
3/7
Vectors to Solve Navigational Problems
Quiz: Basic Vector Operations, Geometric Vectors and
Triangle Apps
Page 18 (Bottom)
Review
Study!
Thursday
3/2
Friday
3/3
Wednesday
3/8
Thursday
3/9
Unit 10 Test
1
Notes on Basic Vector Operations
A Vector Quantity is something that has direction and magnitude (size). An example is velocity, because when
you are traveling, it is important to know direction as well as your speed (therefore, velocity is composed of
direction and speed).
A Scalar Quantity has no direction, such as volume or speed.
What does a vector look like?
What exactly is a vector?
A vector is a _______________ line segment. Two vectors are equal if they have the ____________ length and
go in the ____________ direction. The ________________ of a vector is its length.
Component Form of a vector.
The component form of a vector is used to help simplify
operations involving vectors.
The component form of the vector with initial point
P = ( p1, p2 ) and terminating point Q = (q1,q2 )
is given by:
v
PQ  q1  p1, q2  p2  v
Write the following in Component Form:
1. v
3. Initial Point: (2, -5)
2. a
Terminal Point: (-1, 8)
2
Vector Operations: Because Vectors have specific length and direction, we can perform algebraic operations to
them.



For the following: let u = <-3, 4> , v = <2, 5>, and w =<0, -3>. Find the following and sketch the resultant
vectors.
 
1. u + w


2. 2 v --3 u
3.
1 1 
v w
2
3
Vectors can also be written in, what we call standard unit vectors. In other words, the vector in terms of its’ unit
length in a direction.
In the x-direction, the standard unit (one space) is _____
In the y-direction, the standard unit (one space) is ______
Therefore, a standard unit vector could be written as ________________
Examples using the standard unit vector:
3
Magnitude of a vector: How do we find the length of a vector?
Find the magnitude of the following vectors:
1. <4, 3>
2. <-2, 5>
3. i – 3j
How to find a unit vector: After we have found the magnitude of a vector, we can find the “unit” vector. That is
a vector that is in the same direction of the original vector but only has a length of one unit.
Find the unit vectors for the following:


1. Let u =<3, 4>

2. u =<5, -12>

3. w =-2i + 5j

4. Find the vector, v , with the given magnitude and the same direction as u .

v 6

u  2,5 
4


Let the graph below represent vectors v and u . Perform the following operations and sketch a graph of the
specified vector.

1. -2 v
 
2. v + u


3. u - 2 v
5
Notes on Basic Vector Operations (day 2)
Review from yesterday.
How would we write a vector with its’ initial point at (-3, 5) and its’ terminal point at (5, -1) in:
a. Component form:
b. Standard unit form:
Change the following vectors from component to standard unit form.
a. <-3, 9>
b. <6, 0>
Change the following vectors from standard unit to component form.



1
2
b. v   (4i  8 j )
a. v  2i  6 j





c. Let v  2i  3 j and u  3i  j . Find w if w  2v  u
How to find the magnitude and direction of a vector.
Find the magnitudes and direction of the following vectors.

1. v = <2, 2>

*2. v = <-3, 4>
6
Trigonometry and Vectors:
If we know the angle at which a vector is headed, we use right triangles to write the vector is Standard Unit Form.
x-coordinate: _____
y-coordinate: _______
The vector in Standard Unit Form: ____________
Write the vectors in Standard Unit.
1.
2.
u  6cm
w  10cm
w
u
75
135
3.
v
60
v  4cm
Sum of Vectors:
Use the above to find the magnitude and direction of each resultant vector.
1. v  w
v  w  _____
  ______
2.
wu
w  u  ______
  _______
3.
2w  u
2 w  u  _____
  _______
7
Find the vectors magnitude and direction:

1. v  2(cos 30i  sin 30 j )

2. v  3i  3i

Find the component form of v given its magnitude and angle with the positive x-axis.

1. v  6   45

2. v  3 2   150

3. v  6 and is in the direction of 2i  6 j
8
Notes on Geometric and Algebraic Vectors
Geometric vectors: Because vectors have both magnitude (Length) and direction, we can use them to solve all
sorts of Geometric problems:


1. To add Vectors, use head to tail method.
2. Because vectors are directional v and  v are going in
opposite
directions.
Look at the parallelogram below. Given:
BC  x
CD  y
and P is the midpoint of AC and BD , find each of
the following in terms of x and y .
B
C
P
A
1. AD =
6. BP =
In ∆ABC,
D
2. AB =
7. CA =
3. BA =
4. BD =
5. CB =
8. PA =
9. PC =
AP 3
PQ 1
 and
 . If PQ  x and QB  y , express the following in terms of x and y .
PB 1
QC 4
A
10. BP =
11. PC =
P
12. BC =
13. AP =
14. PA =
15. AB =
Q
B
16. AC =
C
17. BA =
9
Algebraic Vectors: Vectors must follow all algebraic properties, as you will see in the following examples.
1. If v  1,4 and u  3,6 find the following:
v
v
v
b.  v  w (find direction also)
a. v  2w
2. Find the value of “k” that will make the vectors parallel and Perpendicular: 3,2 , 5, k
3. If AB = 5, 3 and A = 2,5 , The Find B.
4. Find the Coordinate of point P if A = (-2, 4) and B = (7, 16) and P is
5. Solve the following Vector equations:
a.  3,4    x, y  4,2 

2
of the way from A to B.
3
b.  2,5  3  x, y  2,7 

*6. let v  2,75  Write v in component form.
10
Assignment Geometric and Algebraic Vectors
Show all work.
I. Look at the parallelogram below. Given:
AB  x
AD  y
the following in terms of x and y .
and P is the midpoint of AC and BD , find each of
B
C
P
A
D
1.
BC =
2.
AB =
3.
DC =
4.
DB =
5.
CB =
6.
BP =
7.
CA =
8.
PA =
9.
PC =
II:
In ∆ABC,
AP 2
PQ 1
 and
 . If PQ  v and QB  w , express the following in terms of
PB 1
QC 3
v and w .
A
P
Q
C
B
10.
BP =
11.
PC =
12.
BC =
13.
AP =
14.
PA =
15.
BA =
16.
AC =
17.
AB =
11
Determine the value of k for which each pair of vectors is parallel and the value of k for which the vectors are
perpendicular.
18) (2, 9 ), (4, k)
19) (3, 10), (5, k)
20)(k, 1), (k, -2)
24) Find the requested point.
a) If AB = (2,3), and A = (3,-2), find B.
B.
21)(13, 8), (k, 2)
25) Find the coordinates of point P
a) A = (0,0), B = (6,3), and P is 1/3 of the way from A to
b) If AB = (-3,4) and B = (-1,2), find A.
b) A = (1,4), B = (5,-4), and P is ¼ of the way from A to
c) If AB = (-1,-1) and A = (0, 5), find B.
c)A = (7,-2), B = (2,8), and P is 4/5 of the way from A to
B.
B.
Solve the following vector equations.
26.  0,3    x, y  3,5 
27.  2  1,3    x, y  0,0 
28.  3,1  2  x, y  4  1,2 
29.   2,9  3  x, y  1,2 
Review:
30. Let v = (2,3) and w = (1,2), find the magnitude and direction of the following: (round to three decimal place for
direction)
a) v
b) v + w
c) 5v -3w


31. Use the fact that: v  3,40  meaning that v has a magnitude of 3 and a direction of 40 off the x-axis
to find the following:

a. write v in Standard Unit Form.

b. Write v in component form.

c. write 2 v in component form.
12
Notes on Dot Product
Dot Product
Sometimes called the “scalar” product because it multiplies two vectors together and gives a “single”
scalar as its product.
Dot Product: Let A  a1,a2 and B  b1,b2 , then the Dot Product is defined as…
A  B  a1 b1  a2 b2


Examples: Find the Dot Products of each pair of vectors. Let u  2,4  v  3,1 

w  3i  4 j
1. u  v Is this a scalar or a vector?
2.
3u  v Is this a scalar or a vector?
3. w v
4.
u  v  w

Is this a scalar or a vector?

4. u  3, 6  v  2 , 6  Find u  v .
Dot Product and Magnitude:
Rule:
2  
u  u u

Use the Dot Product of u  2,4  to find the magnitude.
One of the main uses for the Dot Product is to find the measure of the angle between two vectors.
13
Measure of the angle between two vectors.


1  A  B 
cos
To find the angle between two non-zero vectors:
 A B 


Graph the vectors, and find the angle formed between the following.
1. <-2, 5> and <1, 4>

u  2i  3 j
2. 
v i2j
3. A triangle is formed by the points (-3, -4), (1, 7), and (8, 2). Find the interior angles of the triangle.
4. Find
 
u v
given the angle between them:
u  100, v  250,  

6
.
14
Parallel and Orthogonal Vectors:
Parallel
Orthogonal
*Two vectors are parallel when their slopes are the same.
*Two vectors are orthogonal when their dot product = 0.
Determine if the vectors are Parallel, Orthogonal, or neither.
1.
u  3,5
v  6, 10

u  3i  2 j
2. 
v  4i  6 j
Projection: Projection is the part (or projection) that one vector places on another.
u v 


v
To find projection, use the formula: proj v u 
. This formula shows the part of u that projects onto
2
 v 



v.
Find the projection of

u
onto


v , then write u
as the sum of two orthogonal vectors, one of which is
projv u .


u =<4,2> and v =<1, -2>
15
Notes on Vector Applications
Navigation Examples
Ex. A ship sails 100 km east, followed by 40 km along a bearing of 120°. How far is the ship from its starting
point? What is the bearing of the ship from its starting point?
Ex. An airplane flies 240 miles on a bearing of 25° and then turns and flies 160 miles along a bearing of 130°. How
far is the plane from its starting point? What is the bearing of the plane from its starting point?
Ex. An airplane is traveling in the direction 200° with an airspeed of 250 mi/h. There is a 35 mi-per-hour wind
with a direction 285°. Find the plane’s ground speed and course.
Ex. A plane is traveling in the direction 160° with an airspeed of 400 mi/h. Its course and ground speed are 145°
and 385 mi/h respectively. What is the direction and speed of the wind?
16
Assignment on Vector Applications
Work the following on notebook paper. Give decimal answers correct to three decimal places.
1. Forces of34 pounds and 46 pounds make an angle of 42° with each other and are applied to an
object at the same point. Find the magnitude of the resultant force.
2. Joe Jamoke and Ivan Hoe are pulling up a tree stump. Joe can pull with a force of200 pounds and
Ivan with a force of250 pounds. A total force of 400 pounds is sufficient to pull up the stump.
(a) If they pull at an angle of25° to each other, will the sum of their force vectors be enough to
pull up the stump? What is the sum of their force vectors?
(b) At what angle must they pull in order to exert exactly the 400 pounds needed to pull up the
stump?
·
Freda
3. Freda Pulliam and Yank Hardy are on opposite sides of
a canal, pulling a barge with tow ropes. Freda exerts a
force of 50 pounds at an angle of 20° to the canal , and
Yank pulls at an angle of 15° with just enough force so
that the resultant force vector is directly along the
canal. Find the number of pounds with which Yank
must pull and the magnitude of the resultant vector.
Ya n k
4. Aloha Airlines Flight 007 is approaching Kahului Airport at
an altitude of 5 km. The angle of depression from the plane
to the airport is 9° 32'.
·
(a) What is the plane 's ground distance from the airport?
(b) If the plane descends directly along the line of sight, how
far will it travel along this line in reaching the airport?
5. An airplane has an airspeed of 450 mi/h and a heading of 110°. The wind is blowing
from the east at 23 mi/h. Find the ground speed of the plane and its course.
6. A boat travels at 15 mi/h on a compass heading of 200°. The velocity of the current is
3 m/h toward the north. Find the speed of the boat relative to land, and find its course.
17
.
7. An airplane is flying through the air at a speed of 500 km/h. At the same time, the air is moving with
respect to the ground at an angle of 23° to the plane's path through the air with a speed of 40 km/h
(i.e., the wind speed is 40 km/h). The plane 's ground speed is the magnitude of the vector sum of the
plane 's velocity and the air's velocity with respect to the ground. Find
the plane's ground speed if it is flying:
(a) Against the wind (b) With the wind
8. A ship is sailing through the water in the English Channel with a
velocity of 22 knots along a bearing of 157°, as shown in the figure. (A
knot is a nautical mile per hour, or slightly faster than a regular mile per
hour .) The current has a velocity of 5 knots along a bearing of 213°.
The actual velocity of the ship is the vector sum of the ship's velocity
and the water's velocity. Find the actual velocity.
9. A navigator on an airplane knows that the plane 's velocity through the air
is 250 km/h on a bearing of 237°. By observing the motion of the plane's
shadow across the ground, she finds to her surprise that the plane 's ground speed is only 52 km/h, and
its direction is along a bearing of 15°. She realizes that ground velocity is the vector sum of the plane 's
air velocity and the velocity of the wind. What wind velocity would account for the observed ground
velocity?
Assignment on Vector Applications (Day 2)
1. A ship steams 100 miles east, and then 40 miles on a heading of 120°. How far is the ship and how does
it bear from its starting point?
2. A ship sails 150 miles on a heading of 220° and then turns and sails directly east .for 50 miles. How far is the ship and how
does it bear from its starting point?
3. An airplane flies on a compass heading of 90° at 200 miles per hour. The wind affecting the plane is blowing from 300° at 30
miles per hour. What is the true course and ground speed of the airplane?
4. Let the airplane in Exercise 3 fly 250 miles per hour on a heading of 180°. If the wind direction and speed are
the same as given , what are the true course and ground speed of the airplane?
5. At what compass heading and air speed should an aircraft fly if a wind of 40 miles per hour is blowing from the north, and
the pilot wants to maintain a ground speed of 200 miles per hour on a true course of 90°?
6. Let the wind in Exercise 5 be blowing at 40 miles per hour from 305°, while the pilot still maintains the same
true course and ground speed. What should be his compass heading and air speed?
7. A ship is moving through the water on a compass heading of 30° at a speed of 20 knots (nautical miles per
hour). It is traveling in a current that causes the ship to move on a path with a heading of 45 . Find the speed
of the current if it is flowing directly from the north .
8. A plane is flying with a compass heading of 300 at an air speed of 300 miles per hour. If its true course is observed to be
3 3 0 ° , and if the wind is blowing from 245°, what is the speed of the wind?
9. Two ships leave a harbor, one traveling at 20 knots on a course of 80° and the other at 24 knots on a course of 140°. How far
apart are the ships after two hours? What is the bearing from the first ship to the second at that time?
10.
Two airplanes leave an airport at noon. one flying on a true course of 345°: and the other on a true course of
45°. If the first airplane averages 240 miles per hour ground speed and the other 200 miles per h our ground speed , how far
apart are the airplanes after one hour ? How does the second ai rplane bear from the first ?
11.
A pilot makes a flight plan that will take him from city A to city B, a distance of 400 miles. City B bears 60°
from city A. and the wind at the planned flight altitude is 30 miles per hour from 160°. If the airplane cruises at 320
miles per hour air speed, and if the pilot takes off at noon. what will be his compass heading and what is his ETA
(estimated time of arrival) at city B?
12.
When the pilot in Exercise 11 decides to return to city A , the wind has shifted to 40 miles per hour from 90°. What
must be his compass heading for the return trip, and, if he takes off at 6:00P .M ., what will be his ETA at city A ?
18