A vector is a quantity that has both magnitude and
direction. It is represented by an arrow. The length of
the vector represents the magnitude and the arrow
indicates the direction of the vector.
Blue and orange
vectors have
same magnitude
but different
direction.
Blue and purple
vectors have
same magnitude
and direction so
they are equal.
Blue and green
vectors have
same direction
but different
magnitude.
Two vectors are equal if they have the same direction and
magnitude (length).
Vectors are often used to represent forces of varying
magnitudes and angles. This figure shows 2
baseballs hit at two different angles. The length of
the vector (magnitude) is the speed the baseball is
hit.
How can we find the magnitude if we
have the initial point and the terminal
point? The distance formula
Q
x2 , y2 
Terminal
Point
Initial
Point
x1, y1 
P
How can we find the direction? (Is this all looking familiar
for each application? You can make a right triangle and
use trig to get the angle!)
Although it is possible to do this for
any initial and terminal points, since
vectors are equal as long as the
direction and magnitude are the
same, it is easiest to find a vector
with initial point at the origin and
terminal point (x, y).
Initial
Point
0x1,, 0y1
P
Q
xx,
2 , yy
2
Terminal
Point
A vector whose
initial point is
the origin is
called a
position vector
If we subtract the initial point from the
terminal point, we will have an
equivalent vector with initial point at
the origin.
To
Toadd
addvectors,
vectors,we
weput
putthe
theinitial
initialpoint
pointof
ofthe
thesecond
second
vector
vector on
onthe
theterminal
terminalpoint
pointof
ofthe
thefirst
firstvector.
vector. The
The
resultant
resultantvector
vector has
hasan
aninitial
initialpoint
pointat
atthe
theinitial
initialpoint
point
of
ofthe
thefirst
firstvector
vector and
andaaterminal
terminalpoint
pointat
atthe
theterminal
terminal
point
pointof
ofthe
thesecond
secondvector
vector(see
(seebelow--better
below--bettershown
shown
than
thanput
putin
inwords).
words).
Terminal
point of w
vw
Initial point of V
v
w
w
Move W over keeping
the magnitude and
direction the same.
Vectors are denoted with bold
capital letters
A  a1 , a2
A  a1 , a2  a1i  a2 j
(a1, a2)
j
i
v  3, 2
This is the notation for a
position vector. This means
the point (a1, a2) is the
terminal point and the initial
point is the origin.
We use vectors that are only 1
unit long to build position
vectors. i is a vector 1 unit long
in the x direction and j is a vector
1 unit long in the y direction.
(3, 2)
j
j
i i i
v  3i  2 j
The negative of a vector is just a vector going the opposite
way.
v
v
A number multiplied in front of a vector is called a scalar. It
means to take the vector and add together that many times.
3v
v
v
v
v
u
w
Using the vectors shown,
find the following:
uv
 3w
w
w
w
uv
u
2u  3w  v v
u
u
u
v
w
w
w
v
If we want to add vectors that are in the form ai + bj, we can
just add the i components and then the j components.
v  2i  5 j
w  3i  4 j
v  w   2i  5 j  3i  4 j  i  j
Let's look at this geometrically:
Can you
see from
this picture
5j
how to find
the length
of v?
3i
w
v
 2i i
 4j
j
When we want to know
the magnitude of the
vector (remember this is
the length) we denote it
V

 2  5
2
 29
2
A unit vector is a vector with magnitude 1.
If we want to find the unit vector having the same
direction as a given vector, we find the magnitude of the
vector and divide the vector by that value.
w  3i  4 j
w
3   4
2
What is w ?
2
 25  5
If we want to find the unit vector having the same direction
as w we need to divide w by 5.
3 4
u i j
5 5
Let's check this to see if it really is
1 unit long.
2
2
25
3  4
u      
1
25
5  5
If we know the magnitude and direction of the vector, let's
see if we can express the vector in ai + bj form.
v  5,   150
5
As usual we can use the trig
we know to find the length
in the horizontal direction
and in the vertical direction.
150
v  v  cos  i  sin  j
5 3
5
v  5cos150i  sin 150 j  
i j
2
2
RULES FOR WORKING WITH VECTORS
If A  a1 , a2 , B  b1 ,b2 ,
and k is a scalar, then
1. kA  ka1 , ka2
Scalar product
2. A  B  a1  b1 , a2  b2 Vector sum
This
youahave
a scalar (number) in front of a
3. Asays
 Bif 
1  b1 , a2  b2 Vector difference
vector you can distribute it to each component.
4. A • B  a1b1  a2b2
Dot product
If A  a1 , a2 , B  b1 ,b2 ,
If
A

a
,
a
,
B

b
,b
,
1
2
1
2
and k is a scalar, then
and k is a scalar, then
1. kA  ka1 , ka2
Scalar product
1. kA  ka1 , ka2
Scalar product
2. A  B  a1  b1 , a2  b2 Vector sum
2. A  B  a1  b1 , a2  b2 Vector sum
3. A  B  a1  b1 , a2  b2 Vector difference
ThisAsays
if
youawant
to, a
addtwo
vectors,
you
just add
3.

B

b
b
Vector
difference
4.
• B  a b1 components.
a1 b 2 2 Dot product
the A
corresponding
1 1
2 2
4. A • B  a1b1  a2b2
Dot product
If A  a1 , a2 , B  b1 ,b2 ,
and k is a scalar, then
If A  a1 , a2 , B  b1 ,b2 ,
1. kA  ka1 , ka2
Scalar product
and k is a scalar, then
2. A  B  a1  b1 , a2  b2 Vector sum
1. kA  ka1 , ka2
Scalar product
3. A  B  a1  b1 , a2  b2 Vector difference
2. A  B  a1  b1 , a2  b2 Vector sum
4. A • B  a1b1  a2b2
Dot product
ThisAsays
if
youawant
to, a
subtract
two
vectors,
you just
3.

B

b

b
Vector
difference
1
1
2 components.
2
subtract the corresponding
4. A • B  a1b1  a2b2
Dot product
The definition of the product of two vectors is:
A  B  a1b1  a2b2
where A  a1i  a2 j and B  b1i  b2 j
If v  2i  5 j and w  4i 1 j, find v  w
v  w    2  4   5 1
 8  5  3
This is called the dot product. Notice the answer is just
a number NOT a vector.
Summary of Vector Rules
If A  a1 , a2 , B  b1 ,b2 ,
and k is a scalar, then
1. kA  ka1 , ka2
Scalar product
2. A  B  a1  b1 , a2  b2 Vector sum
3. A  B  a1  b1 , a2  b2 Vector difference
4. A • B  a1b1  a2b2
Dot product
The dot product is useful for several things. One of
the important uses is in a formula for finding the angle
between two vectors that have the same initial point.
If u and v are two nonzero vectors, the angle
 , 0   <  , between u and v is determined
by the formula
v

u
uv
cos  
u v
Technically there are two angles between
these vectors, one going the "shortest" way
and one going around the other way. We
are talking about the smaller of the two.
Find the angle  between u = 2i  j and v = 4i + 3 j.
uv
cos  
u v
u  v   2 4    1 3  8  3  5
u  2   1  5
2
2
v  4  3  16  9  25  5
2
v  4i  3j

u  2i  j
2
uv
5
1
cos  


u
5 5 5
  cos
1
1
 63.4
5
Find the angle between the vectors
v = 3i + 2j and w = 6i + 4j
18  8
vw


cos  
v w
13 52
  cos 1  0
1
w  6i  4 j
v  3i  2 j
26
676
1
What does it mean when the
angle between the vectors is 0?
The vectors have the same
direction. We say they are
parallel because remember
vectors can be moved around
as long as you don't change
magnitude or direction.

If the angle between 2 vectors is
, what would their dot
w
=
2i
+
8j
2
product be?

uv

cos  
u v
2
Since cos 2 is 0, the
dot product must be 0.
Vectors u and v in thisvcase
are- called
orthogonal.
=
4i
j
(similar to perpendicular but refers to vectors).
Determine whether the vectors v = 4i - j and
w = 2i + 8j are orthogonal.
compute their dot product
and see if it is 0
v  w   4 2    1 8  0
v w  0
The vectors v and w are orthogonal.