ectors
The word
vector
comes from the Latin word
vectus
which means “carried”. It is best
to think of a vector as the displacement from an
initial point
P
to a
terminal point
Q
.
Such a vector is expressed as
. The figure below shows a typical vector. Notice that
the vector appears as a line with an arrow at one end. This is referred to as a directed line
segment. The end with the arrow is the head of the vector, or the terminal point, while the
other end is the starting, or initial point.
PQ
Vectors have two important aspects: direction and magnitude. The
direction tells you where to point the vector and the magnitude tells
you how far to go in that direction.
The magnitude of the vector
PQ
is written as
PQ
. Note that while
this looks just like we take the absolute value of the vector
, it
means the distance between points
P
and
Q
. And so, to compute the
magnitude of a vector, we use the distance formula. We use two
different formulas, depending on whether or not the vector is in two
dimensions or three dimensions.
PQ
P
Q
If our points are in two dimensions, then
P
can be expressed as (
p
1
,
p
2
) and
Q
can be
expressed as (
q
1
,
q
2
). We have that
PQ
can be written as
11
2 2
,
qp
and
22
()
(
PQ
q
p
q
p
)
1
.
It helps to think of the magnitude of a vector as the hypotenuse of a triangle. Let us
consider the above vector placed on the coordinate axes.
Notice that the “base” of the triangle (which also happens
to be parallel to the
x
axis) is
1
while the “height” of the
triangle (parallel to the
y
axis) is
2
q
P
Q
p
1
q
1
2
2
2
p
. To find the length of
the hypotenuse, we would square the two sides, add them
together, and take the square root, just as we did above.
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 Spring '07
 Hohnhold
 Calculus, Vectors, Vector Space

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