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Section 8.4 Vectors
In this section, we study vectors. Vectors are very useful mathematical tools, and they are especially
important for the study of physical applications of mathematics.
A
vector
, denoted in these notes in boldface, like
v
, is a quantity that has both
magnitude and
direction
. Vectors are frequently represented by arrows. When a vector
v
is represented by an
arrow, the length of the arrow indicates the
magnitude
and the direction of the arrow indicates
the direction of the vector. The magnitude of
v
is denoted by

v

.
If
v
is a vector and

v

=1
,then
v
is called a
unit vector
.A
position vector
is a vector that
has its initial point (tail) at the origin. The
zero vector
, indicated by
0
, is the vector having
magnitude equal to zero. Two vectors are
equal
if they have equal magnitude and direction. If
u
and
v
are equal, we write
u=v
. We do not distinguish between translates of vectors; hence, two
parallel vectors of equal length pointing in the same direction are equal.
We give some examples to show the diﬀerence between vectors and
scalars
(numbers). The number
3 has magnitude but not direction. Hence, the number 3 is a scalar and not a vector. The velocity
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 Fall '11
 Kutter
 Algebra, Vectors

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