We’ll start the chapter off with a fairly short discussion
introducing the 3-D coordinate system and the conventions that we’ll be
using. We will also take a brief look at
how the different coordinate systems can change the graph of an equation.
Let’s first get some basic notation out of the way. The 3-D coordinate system is often denoted by
. Likewise the 2-D coordinate system is often
denoted by and the 1-D coordinate system is denoted by . Also, as you might have guessed then a
general n dimensional coordinate
system is often denoted by .
Next, let’s take a quick look at the basic coordinate
This is the standard placement of the axes in this
class. It is assumed that only the
positive directions are shown by the axes.
If we need the negative axes for any reason we will put them in as
Also note the various points on this sketch. The point P
is the general point sitting out in 3-D space.
If we start at P and drop
straight down until we reach a z-coordinate
of zero we arrive at the point Q. We say that Q sits in the xy-plane. The xy-plane
corresponds to all the points which have a zero z-coordinate. We can also
start at P and move in the other two
directions as shown to get points in the xz-plane
(this is S with a y-coordinate of zero) and the yz-plane (this is R with an x-coordinate of
Collectively, the xy,
xz, and yz-planes are sometimes called the coordinate planes. In the remainder of this class you will need
to be able to deal with the various coordinate planes so make sure that you
Also, the point Q
is often referred to as the projection of P
in the xy-plane. Likewise, R
is the projection of P in the yz-plane and S is the projection of P
in the xz-plane.
Many of the formulas that you are used to working with in have natural extensions in . For instance the distance between two points
in is given by,
While the distance between any two points in is given by,
Likewise, the general equation for a circle with center and radius r
is given by,
and the general equation for a sphere with center and radius r
is given by,
With that said we do need to be careful about just
translating everything we know about into and assuming that it will work the same
way. A good example of this is in
graphing to some extent. Consider the
Note that at this point we can now write down the equations
for each of the coordinate planes as well using this idea.
Let’s take a look at a slightly more general example.
Example 2 Graph
in and .
Of course we had to throw out for this example since there are two
variables which means that we can’t be in a 1-D space.
In this is a line with slope 2 and a y intercept of -3.
However, in this is not necessarily a line. Because we have not specified a value of z we are forced to let z take any value. This means that at any particular value of z we will get a copy of this
line. So, the graph is then a vertical
plane that lies over the line given by in the xy-plane.
Here is the graph in .
here is the graph in .
Notice that if we look to where the plane intersects the xy-plane we will get the graph of the
line in as noted in the above graph by the red line
through the plane.
Let’s take a look at one more example of the difference
between graphs in the different coordinate systems.
Example 3 Graph
in and .
As with the previous example this won’t have a 1-D graph
since there are two variables.
In this is a circle centered at the origin with
In however, as with the previous example, this
may or may not be a circle. Since we
have not specified z in any way we
must assume that z can take on any
value. In other words, at any value of
z this equation must be satisfied
and so at any value z we have a
circle of radius 2 centered on the z-axis. This means that we have a cylinder of
radius 2 centered on the z-axis.
Here are the graphs for this example.
Notice that again, if we look to where the cylinder
intersects the xy-plane we will
again get the circle from
We need to be careful with the last two examples. It would be tempting to take the results of
these and say that we can’t graph lines or circles in and yet that doesn’t really make sense. There is no reason for there to not be graphs of lines or circles in . Let’s think about the example of the
circle. To graph a circle in we would need to do something like at . This would be a circle of radius 2 centered
on the z-axis at the level of . So, as long as we specify a z we will get a circle and not a
cylinder. We will see an easier way to specify circles in a later section.
We could do the same thing with the line from the second
example. However, we will be looking at
lines in more generality in the next section and so we’ll see a better way to
deal with lines in there.
The point of the examples in this section is to make sure
that we are being careful with graphing equations and making sure that we
always remember which coordinate system that we are in.
Another quick point to make here is that, as we’ve seen in
the above examples, many graphs of equations in are surfaces.
That doesn’t mean that we can’t graph curves in . We can and will graph curves in as well as we’ll see later in this chapter.