12
Three Dimensions
So far we have been investigating functions of the form
y
=
f
(
x
), with one independent and
one dependent variable. Such functions can be represented in two dimensions, using two
numerical axes that allow us to identify every point in the plane with two numbers. We now
want to talk about threedimensional space; to identify every point in three dimensions
we require three numerical values.
The obvious way to make this association is to add
one new axis, perpendicular to the
x
and
y
axes we already understand.
We could, for
example, add a third axis, the
z
axis, with the positive
z
axis coming straight out of the
page, and the negative
z
axis going out the back of the page. This is difficult to work with
on a printed page, so more often we draw a view of the three axes from an angle:
.
x
y
z
275
276
Chapter 12 Three Dimensions
You must then imagine that the
z
axis is perpendicular to the other two. Just as we have
investigated functions of the form
y
=
f
(
x
) in two dimensions, we will investigate three
dimensions largely by considering functions; now the functions will (typically) have the
form
z
=
f
(
x, y
). Because we are used to having the result of a function graphed in the
vertical direction, it is somewhat easier to maintain that convention in three dimensions.
To accomplish this, we normally rotate the axes so that
z
points up; the result is then:
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•
(2
,
4
,
5)
y
z
x
4
2
5
Note that if you imagine looking down from above, along the
z
axis, the positive
z
axis
will come straight toward you, the positive
y
axis will point up, and the positive
x
axis
will point to your right, as usual. Any point in space is identified by providing the three
coordinates of the point, as shown; naturally, we list the coordinates in the order (
x, y, z
).
One useful way to think of this is to use the
x
and
y
coordinates to identify a point in the
x

y
plane, then move straight up (or down) a distance given by the
z
coordinate.
It is now fairly simple to understand some “shapes” in three dimensions that corre
spond to simple conditions on the coordinates.
In two dimensions the equation
x
= 1
describes the vertical line through (1
,
0). In three dimensions, it still describes all points
with
x
coordinate 1, but this is now a plane, as in figure 12.1.
Recall the very useful distance formula in two dimensions: the distance between points
(
x
1
, y
1
) and (
x
2
, y
2
) is
radicalbig
(
x
1
−
x
2
)
2
+ (
y
1
−
y
2
)
2
; this comes directly from the Pythagorean
theorem.
What is the distance between two points (
x
1
, y
1
, z
1
) and (
x
2
, y
2
, z
2
) in three
dimensions?
Geometrically, we want the length of the long diagonal labeled
c
in the
“box” in figure 12.2.
Since
a
,
b
,
c
form a right triangle,
a
2
+
b
2
=
c
2
.
b
is the vertical
distance between (
x
1
, y
1
, z
1
) and (
x
2
, y
2
, z
2
), so
b
=

z
1
−
z
2

. The length
a
runs parallel
to the
x

y
plane, so it is simply the distance between (
x
1
, y
1
) and (
x
2
, y
2
), that is,
a
2
=
(
x
1
−
x
2
)
2
+ (
y
1
−
y
2
)
2
. Now we see that
c
2
= (
x
1
−
x
2
)
2
+ (
y
1
−
y
2
)
2
+ (
z
1
−
z
2
)
2
and
c
=
radicalbig
(
x
1
−
x
2
)
2
+ (
y
1
−
y
2
)
2
+ (
z
1
−
z
2
)
2
.
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 Fall '11
 GUICHARD
 Vector Space, Cartesian Coordinate System, Dot Product, ... ...

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