(8/1/08)
Math 20C. Lecture Examples.
Section 12.1. Vectors in the plane
†
Definition 1
A
vector
v
represents a nonnegative number and, if the number is not zero, a direction.
The number associated with the vector
v
is called its
length
or
magnitude
and is denoted

v

. The
vector of zero length is the
zero vector
and is denoted
O
. It has no direction.
Vectors are generally denoted by boldfaced letters, like the
v
in Definition 1, or by letters with
arrows over them, as in the symbol
→
PQ
.
Vectors are represented in drawings by arrows, where in each case the length of the arrow is
the magnitude of the vector, measured with a scale that might or might not be the scale used on the
coordinate axes. The direction of the arrow is the direction associated with the vector (Figure 1). The
same vector can be represented by different arrows in different locations (Figure 2), provided that the
different arrows are parallel, have the same lengths, and point in the same direction.
v

v

v
v
FIGURE 1
FIGURE 2
If a vector is represented by an arrow in an
xy
plane as in Figure 3, then the
x
 and
y
components
of the vector are the changes in the
x
 and
y
coordinates from the base to the tip of the arrow, measured
with the scale used for measuring the length of the arrow. If the
x
component of
v
is
a
and the
y
component is
b
, as shown in Figure 6 with positive
a
and
b
, we write
v
=
(
a, b
)
. The length of a
vector can be calculated from its
x
 and
y
components by using the Pythagorean Theorem:

v

=
(
a, b
)
=
radicalbig
a
2
+
b
2
.
x
y
v
=
(
a, b
)
a
b
FIGURE 3
†
Lecture notes to accompany Section 12.1 of
Calculus, Early Transcendentals
by Rogawski.
1
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Math 20C. Lecture Examples. (8/1/08)
Section 12.1, p. 2
A nonzero vector
v
in an
xy
plane can also be described by giving its length

v

and its
angle
of inclination
, which is an angle
θ
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 Spring '08
 Helton
 Calculus, Addition, Vectors, Negative and nonnegative numbers, Angle of inclination

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