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Unformatted text preview: 5E13(pp 828837) 1/18/06 11:09 AM Page 828 CHAPTER 13 Wind velocity is a vector because
it has both magnitude and direction. Pictured are velocity vectors
indicating the wind pattern over
San Francisco Bay at 12:00 P.M.
on June 11, 2002. V ectors and the Geometry of Space 5E13(pp 828837) 1/18/06 11:09 AM Page 829 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 z O
y
x FIGURE 1 Coordinate axes
z y 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 a, b of real numbers, where a is the
xcoordinate and b is the ycoordinate. 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 a, b, c of real numbers.
In order to represent points in space, we ﬁrst choose a ﬁxed point O (the origin) and
three directed lines through O that are perpendicular to each other, called the coordinate
axes and labeled the xaxis, yaxis, and zaxis. Usually we think of the x and yaxes as
being horizontal and the zaxis as being vertical, and we draw the orientation of the axes
as in Figure 1. The direction of the zaxis is determined by the righthand rule as illustrated in Figure 2: If you curl the ﬁngers of your right hand around the zaxis in the direction of a 90 counterclockwise rotation from the positive xaxis to the positive yaxis, then
your thumb points in the positive direction of the zaxis.
The three coordinate axes determine the three coordinate planes illustrated in Figure 3(a). The xyplane is the plane that contains the x and yaxes; the yzplane contains
the y and zaxes; the xzplan...
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