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).
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).
Q
xx,
2 , yy
2
Terminal Point
A vector whose
initial point is the
origin is called a
position vector
Initial Point
0x1,, 0y1
P
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
secondvector
vectoron
onthe
the
terminal
terminalpoint
pointof
ofthe
thefirst
firstvector.
vector. The
Theresultant
resultantvector
vectorhas
hasan
aninitial
initialpoint
point
at
atthe
theinitial
initialpoint
pointof
ofthe
thefirst
firstvector
vectorand
andaaterminal
terminalpoint
pointat
atthe
theterminal
terminal
point
of
the
second
vector
(see
below--better
shown
than
put
in
point of the second vector (see below--better shown than put 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.
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
Using the vectors shown, find the
following:
v
u
w
uv
 3w
w
w
w
uv
u
2u  3w  v v
u
u
u
v
w
w
w
v
Vectors are denoted with bold letters
This is the notation for a position
vector. This means the point (a, b) is
the terminal point and the initial point
is the origin.
a
v   
a
b
v     ai  bj
We use vectors that are only 1 unit long to
b
 
build position vectors. i is a vector 1 unit
(a, b)
long in the x direction and j is a vector 1 unit
long in the y direction.
j
i
 3
v   
 2
(3, 2)
j
j
i i i
v  3i  2 j
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
how to find the
length of v?
3i
w
5j
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