ThreeD Coordinate Systems
John E. Gilbert, Heather Van Ligten, and Benni Goetz
The graph of a function
f
(
x, y
)
of two variables is a surface in
3
-space, and familiar single variable
methods can be used once a coordinate system has been introduced into
3
-space.
The
xyz
-coordinate system in
3
-space is
obtained by adding the
z
-axis perpendicular to the
usual
xy
-coordinate system in the
xy
-plane. Each
point
P
in
3
-space is then determined by a triple
(
a, b, c
)
as shown to the right. The three
coordinate axes
intersect at the origin, and each
pair of axes determines a
coordinate plane
.
If we drop a perpendicular from
P
to the the
xy
-plane, we get a point
Q
with coordinates
(
a, b,
0)
called the
projection
of
P
on the
xy
-plane. In the same way there are projections
R
(0
, b, c
)
and
S
(
a,
0
, c
)
of
P
on the
yz
-plane and
zx
-plane respectively.
x
y
z
P
(
a, b, c
)
a
b
c
The figures below show the three
coordinate planes
from two viewpoints :
x
y
z
4
3
2
One Viewpoint:
x, y, z >
0
x
y
z
4
3
2
-1
Another Viewpoint:
x, z >
0
, y <
0
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Colors help identify the coordinate planes:
xy
-plane
{
(
x, y, z
) :
z
= 0
}
xz
-plane
{
(
x, y, z
) :
y
= 0
}
yz
-plane
{
(
x, y, z
) :
x
= 0
}
It will be important to understand planes parallel to the these coordinate planes.
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- Spring '07
- Sadler
- Multivariable Calculus, coordinate planes, 3-space
-
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