©
:
Pre

Calculus

Chapter 9A
Chapter 9A

A Brief Description
As life becomes more complicated, entities arise that not only have a
magnitude
, but also a
direction
associated with them. Such objects are called
vectors
.
A common example of this occurs when you
give someone directions to get to your home: go 5 blocks west, then 16 blocks north, and my house is
on the corner. There are two vectors here. The first has a magnitude of 5 units and a direction, west;
the second has a magnitude of 16 and a direction, north. These verbal descriptions of a vector are too
cumbersome, and a better notation has been invented. It is the same notation which is used to locate
points in the plane,
R
2
or in threespace,
R
3
.
Representing Vectors
.
Example 1
:
A vector which is one unit long and points due north can be represented by the ordered
pair
⟨
1,0
, and a vector which has magnitude 5 and points
northwest
is
−
5
2
,
5
2
The individual values are called
components
. Since these vectors have only two components, we think
of them as belonging to or lying in a plane. We often picture vectors as arrows. The tail of the arrow is
at the origin and the arrow tip is located at the point whose Cartesian coordinates are the same as the
ordered pair (triple) which represents the vectors. The vectors
⟨
1,1
,
⟨
−
4,3
,and
⟨
3,
−
5
are plotted
below.
5
4
3
2
1
0
4
5

5
4

3

2

1
12345
(1,1)
(4,3)
(3,5)
x
y
Two dimensions
In
R
2
(the plane) there are two special vectors. The two that are parallel to the coordinate axes.
They are commonly denoted by the symbols
i
and
j
or by the pair of symbols
e
1
and
e
2
.Thatis
,
e
1
i
⟨
e
2
j
⟨
0,1
1
0
1
1
1
x
y
1
e
2
e
©
:
Pre

Calculus