Vectors and the Geometry of Space
Wind velocity is a vector because
it has both magnitude and direc
tion.
Pictured are velocity vectors
indicating the wind pattern over
San Francisco Bay at 12:00
P
.
M
.
on June 11, 2002.
C
H A P T E R
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5E13(pp 828837)
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In this chapter we introduce vectors and coordinate systems for
threedimensional space. This will be the setting for our study of
the calculus of functions of two variables in Chapter 15 because
the graph of such a function is a surface in space. In this chapter
we will see that vectors provide particularly simple descriptions
of lines and planes in space.

13.1
ThreeDimensional Coordinate Systems
To locate a point in a plane, two numbers are necessary. We know that any point in
the plane can be represented as an ordered pair
of real numbers, where
is the
coordinate and
is the
coordinate. For this reason, a plane is called twodimensional.
To locate a point in space, three numbers are required. We represent any point in space by
an ordered triple
of real numbers.
In order to represent points in space, we first choose a fixed point
(the origin) and
three directed lines through
that are perpendicular to each other, called the
coordinate
axes
and labeled the axis,
axis, and axis. Usually we think of the  and axes as
being horizontal and the axis as being vertical, and we draw the orientation of the axes
as in Figure 1. The direction of the axis is determined by the
righthand rule
as illus
trated in Figure 2: If you curl the fingers of your right hand around the
axis in the direc
tion of a
counterclockwise rotation from the positive axis to the positive axis, then
your thumb points in the positive direction of the axis.
The three coordinate axes determine the three
coordinate planes
illustrated in Fig
ure 3(a). The
plane is the plane that contains the  and axes; the
plane contains
the  and axes; the
plane contains the  and axes. These three coordinate planes
divide space into eight parts, called
octants
. The
first octant
, in the foreground, is deter
mined by the positive axes.
Because many people have some difficulty visualizing diagrams of threedimensional
figures, you may find it helpful to do the following [see Figure 3(b)]. Look at any bottom
corner of a room and call the corner the origin. The wall on your left is in the
plane, the
wall on your right is in the
plane, and the floor is in the
plane. The axis runs along
the intersection of the floor and the left wall. The
axis runs along the intersection of the
floor and the right wall. The
axis runs up from the floor toward the ceiling along the inter
section of the two walls. You are situated in the first octant, and you can now imagine seven
other rooms situated in the other seven octants (three on the same floor and four on the
floor below), all connected by the common corner point
.
O
z
y
x
xy
y
z
x
z
FIGURE 3
(a) Coordinate planes
y
z
x
O
yz
plane
xy
plane
xz
plane
(b)
z
O
right wall
left wall
y
x
floor
z
x
x
z
z
y
y
z
y
x
xy
z
y
x
90
z
z
z
y
x
z
y
x
O
O
a
,
b
,
c
y
b
x
a
a
,
b
829
FIGURE 2
Righthand rule
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 Fall '09
 hamrick
 Linear Algebra, Vectors, Vector Space, Dot Product, (pp 828837)

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