Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
Think of traveling from point A to point B along AB . This quantity possesses both a magnitude
and a direction. In science, force, velocity and displacement require descriptions involving both
magnitude and direction. These are called vectors. Geometrically, we can think of a vector as
a directed line segment.
The vector shown at the right has an initial point A and a terminal point B.
It can be denoted as v or AB . The direction of the arrowhead indicates
the direction of the vector, and the length of the segment AB indicates its
magnitude, denoted by v .
A
v
B
If A and B are different points, the vector from A to B, AB , always has a positive magnitude. If
A=B, then AA is called the zero vector and is denoted by 0 , which is simply a point.
Two vectors are equal if and only if they have the same direction and the
same magnitude. Consider the four vectors drawn at the right. Vectors a and
b are equal since they have the same direction and same magnitude. Vectors
c and d have the same magnitude but are not equal because they have
opposite direction. We can say that c  d or c  d .
If you travel first from A to B and then from B to C, the result is a
movement from A to C. This leads to the idea of vector addition.
Vector addition involves both direction and magnitude. The sum
of two or more vectors is called the resultant of the vectors. The
vectors making up the sum are called the components. You can
graphically find the resultant by using the triangle method or the
parallelogram method as described below.
a
c
d
C
B
A
Triangle Method: To find the sum of vectors x and y , place the
x y
initial point of y at the terminal point of x . The resultant, x  y , is the
dotted vector shown – its initial point is the initial point of x , and its
terminal point is the terminal point of y .
x
Parallelogram Method: To find the sum of vectors x and y ,
position x and y so that they share the same initial point. Since the
opposite sides of a parallelogram are equal, complete a
parallelogram using x and y as sides and the resultant vector is the
diagonal with the same initial point as the two component vectors.
b
y
x y
x
y
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
The triangle method is especially useful when you are adding more than two vectors. You add
them tip-to-tail in the order of the expression. Using the vectors below, add u  v  w
v
v
w
w
u
u
The difference of two vectors, u and v ,
denoted by u  v , is simply the sum, u  v ,
 
uvw
v
v
u
as illustrated to the right.
v
u v
Equivalently, using the parallelogram method, the diagram below illustrates u  v .
v
u
u v
v
u
v
The operation of vector addition has the following four properties for all vectors, x, y, & z .
1)
Commutative
x y  yx
2)
Associative
 x  y  z  x   y  z
3)
Identity
x0  x
4)
Inverse
x  x  0
 
u
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
Problem Set 1
1) Given vector x , suppose y is a vector such that x  y  x . What can you conclude about y ?
2) Using the vector x , illustrated at the right,
a) draw a vector that has the same direction as x , but has three times the
magnitude (length).
x
b) draw a vector that has the opposite direction from x and three times the magnitude.
3) Given the regular hexagon ABCDEF, let u  AB, v  BC , w  CD, x  DE , y  EF , z  FA .
Name the vectors equal to the sums given below.
A
B
____ a)
u  y   w
____ c) u  x


____ b) x  y  z
C
F
____ d) v  y
____ e) w  z
E
D
4) Simplify the following expression using the vector properties:
_____ a) x  y  0
 
_____ c) x   x  y
_____ b) 0  x  0  y
 
 
 
_____ d) x   y  z   x  y  u   z
5) Given vectors u , v,  u ,  v , label the sets of two vectors and their resultant vector as sums
or differences of the vectors.
v

u
u
v
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
6) Given parallelogram GRAM with BC and DE connecting midpoints of opposite sides, x  OE ,
and y  OC . Write the vectors in terms of x and y .
____ b) DB
____ f) GB
____ j) OG
____ n) OR
____ a) OD
____ e) BE
____ i) GR
____ m) EC
____ c) BR
____ g) CA
____ k) OM
____ o) RA
____ d) CB
____ h) DC
____ l) RE
C
M
A
y
D
O
G
B
E
x
R
7) Consider a rectangular prism with congruent rectangular bases ABCD and EFGH, such that
AE  AB . Also, as labeled below, x  AB, y  AE & z  AD . Name vectors equal to the
sums or differences.
F
E
y
H
z
D
A
G
B
x
C
____a) x  y
____b) x  z
____c) y  z
____d) x  y  z
____e) y  x
____f) z  x
____g) y  z
____h) y  x  z
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
If x is a vector, the sum of x and x is a vector having the same direction as x but whose length
is twice that of x . This is denoted 2 x . Similarly, the sum of x and 2 x is denoted as 3 x , and
so forth. This leads to the consideration of vectors of the form a x where a is a real number,
called a scalar. This combination of numbers with vectors is referred to as scalar
multiplication.
For each real number a, the vector a x , is called a scalar multiple of x .
Problem Set 2
cx
ax
bx
x
dx
1) In the diagram at the left, ax , bx , cx , d x are scalar
multiples of vector x whose initial point is at the origin.
State the scalars:
a =_____
b =_____
c =_____
d =_____
2) Consider the parallelogram, ABCD with diagonals AC and BD . DAE is a straight line. Write
the simplest vector in terms of u , v , w or y or their opposites for each expression.
_____ a) MC
_____ i) AB  DA
_____ b) BC
_____ j) w  y
_____ c) CD
C
D
u
M
w
A
_____ k) AD  AE
y
v
B
u
_____ d) DM
_____ l) v  u  2w
_____ e) AC
_____ m) AE  EB  BM  MC  CD
_____ f) EB
_____ n) 2 y  w  u
_____ g) AB  BC
_____ o) 2w  u  2 y
E
u  AD
w  AM
u  AE y  BM
_____ h) AD  AB
v  AB
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
Vector Diagrams
You can solve problems involving forces, velocity and displacements by drawing vector
diagrams and using your knowledge of trigonometry to solve the problems.
Example: A plane flying due east at 100 m/sec is blown due south at 40 m/sec by a strong
wind. Find the plane’s resultant velocity vector (speed and direction).
Solution:
Begin by sketching a picture - draw the component vectors
Draw in the resultant vector and use your knowledge of trigonometric functions to find the
magnitude of the vector and the angles needed.
100

40
resultant
resultant 2  1002  402
 10000  1600
 11600
resultant  107.703m/sec
40
107.703
 40 
  sin 1 

 107.703 
sin =
Therefore, the resultant vector of the plane is 21.801
South of East at a speed of 107.703 m/sec
  21.801
Problem Set 3
1) An airplane flies due west at 240 km/hr. At the same time, the wind blows it due south at
70 km/hr. Find the plane’s resultant velocity.
2) A hiker leaves camp and walks 15 km due north. The hiker then walks 15 km due east.
What is the hiker’s direction and displacement from the starting point?
3) Two soccer players kick a ball at the same time. One player’s foot exerts a force of 70
newtons west. The other player’s foot exerts a force of 50 newtons north. What is the
magnitude and direction of the resultant force on the ball?
4) An airplane flies at 150 km/hr and heads 30 south of east. A 40 km/hr wind blows it in
the direction 30 west of south. Find the plane’s resultant velocity.
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
The previous sets of problems were easily solved using right triangles. If a right triangle is not
formed, then it becomes necessary to use either the Law of Sines or the Law of Cosines to find
the missing parts of the triangle.
sin A sin B sin C


a
b
c
Recall:
a 2  b 2  c 2  2bc cos A
b 2  a 2  c 2  2ac cos B
c 2  a 2  b 2  2ab cos C
One of the most common applications of vectors is navigation. For navigation, it is best to use
the notion of a bearing. A bearing is a direction angle measured clockwise from due north. In
discussing the direction in which a craft is headed, use the term heading in place of bearing.
An example is given below:
Example: A ship steams due east from a harbor. After proceeding for 120 miles on this course,
it changes its heading to 140 and steams another 70 miles. How far is the ship from the harbor
and what is its bearing from the harbor to the ship?
Solution: It is important to sketch an accurate diagram before proceeding.
N
harbor
Begin by finding the magnitude of r, using
the Law of Cosines:
N

r 2  1202  702  2 120  70  cos130
140
120
B
r
r 2  14400  4900  16800  .643
70
It is possible to find the angle between the
two vectors (B) by subtracting 180  140  40
and adding 90 . Therefore, B  130 . Use angle B
and the magnitudes of the two vectors to find  .
r 2  30098.832
r  173.490 miles
sin  sin130

70 173.490
70sin130
sin  
173.490
 70sin130 
  sin 1 

 173.490 
  18.004
Therefore, the bearing of the ship is 90  18  108 .
The ship is 173.49 miles from harbor on a heading of 108 .
Algebra 2 / Trigonometry
VECTOR PACKET
Name: _______________________
Period: 1 2 3 4 5 6 7
Problem Set 4
1) An airplane is flying 520 mph on a compass heading of 35 .
A wind of 70 mph is blowing from the north. What is the
true course (the resultant vector which is the ground speed
and direction of the airplane)?
2) A ship steams 100 miles east, and then 40 miles on a
heading of 120 . How far is the ship and what is its bearing
from its starting point?
3) A ship steams 45 miles due North, turns to a heading of
45 and steams 25 miles, then turns south and steams
another 40 miles. How far is it from its starting place and
what is its bearing from its starting place?
4) Two forces, one of 100 pounds and the other of 150
pounds, act of the same object, at angles of 20 and 60 ,
respectively with the positive x-axis. Find the direction and
magnitude of the resultant vector of these two forces.
5) Three forces of 75 pounds, 100 pounds, and 125 pounds
act on the same object at angles of 30 , 45 and 120 ,
respectively with the positive x-axis. Find the direction and
magnitude of the resultant vector of these three forces.