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The 3 D Coordinate
System
The 3D Coordinate System
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
...
Let’s first get some basic notation out of the way
...
Likewise the 2-D coordinate system is often denoted by 2 and the 1-D coordinate system
is denoted by
...
Next, let’s take a quick look at the basic coordinate system
...
It is assumed that only the positive
directions are shown by the axes
...
Also note the various points on this sketch
...
If we start at P and drop straight down until we reach a z-coordinate of zero we arrive that
the point Q
...
The xy-plane corresponds to all the points which
have a zero z-coordinate
...
Collectively, the xy, xz, and yz-planes are sometimes called the coordinate planes
...
Also, the point Q is often referred to as the projection of P in the xy-plane
...
Many of the formulas that you are used to working with in 2 have natural extensions in
For instance the distance between two points in 2 is given by,
3
...
A good example of this is in graphing to some
extent
...
Example 1 Graph x = 3 in
2
,
and
3
...
In
2
the equation x = 3 tells us to graph all the points that are in the form ( 3, y )
...
In
3
the equation x = 3 tells us to graph all the points that are in the form ( 3, y, z )
...
So, in a 3-D coordinate system this is a plane
that will be parallel to the yz-plane and pass through the x-axis at x = 3
...
Here is the graph of x = 3 in
2
...
Note that we’ve presented this graph in two different
styles
...
Both views can be convenient on occasion to help with perspective
and so we’ll often do this with 3D graphs and sketches
...
z=0
y=0
xy − plane
xz − plane
x=0
yz − plane
Let’s take a look at a slightly more general example
...
In
2
2
and
3
...
However, in 3 this is not necessarily a line
...
This means that at any particular value of z we will get a copy of
this line
...
Here is the graph in
here is the graph in
2
3
...
Notice that if we look to where the plane intersect the xy-plane we will get the graph of the line in
2
as noted in the above graph by the red line through the plane
...
Example 3 Graph x + y = 4 in
2
2
2
and
3
...
In
2
this is a circle centered at the origin with radius 2
...
Since we have
not specified z in any way we must assume that z can take on any value
...
This means that we have a cylinder of radius 2 centered on the z-axis
...
Notice that again, if we look to where the cylinder intersects the xy-plane we will again get the
circle from 3
...
It would be tempting to take the results of
these and say that we can’t graph lines or circles in 3 and yet that doesn’t really make sense
...
Let’s think about the example
of the circle
...
This would be a circle of radius 2 centered on the z-axis at the level of z = 5
...
We will see an easier way to specify circles in a
later section
...
However, we will be looking
3
at line in more generality in the next section and so we’ll see a better way to deal with lines in
there
...
Another quick point to make here is that, as we’ve seen in the above examples, many graphs of
equations in 3 are surfaces
...
We can and
will graph curves in 3 as well as we’ll see later in this chapter