Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the previous lecture, we saw how to construct a 2-dimensional Cartesian coordinate system
which allows us to graphically display ordered pairs of real numbers
or sets of these ordered pairs as points in a plane.
We did this by taking the Cartesian product of the 1-dimensional number line with itself
to form the 2-dimensional Cartesian plane.
In this lecture, we will construct a 3-dimensional Cartesian coordinate system
which will allow us to display ordered triples of real numbers
as points in 3-dimensional space.
The set of real numbers R
can be thought of as corresponding to a continuum of points in 1-dimensional space.
It is this space which is represented by the number line.
Each real number corresponds to a unique point in this 1-dimensional space.
We then created a 2-dimensional Cartesian coordinate system
by taking the Cartesian product of the set of real numbers with itself
to form an infinite set of ordered pairs called "R squared"
or more commonly "R-two".
R-two corresponds to a continuum of points
in the 2-dimensional space represented by the Cartesian plane
and each ordered pair of real numbers corresponds to a unique point in this 2-dimensional space.
However, we are not limited to forming Cartesian products of only two sets of real numbers.
We can also form the Cartesian product of three sets to form the set "R-three".
Just as the set R-two consists of ordered pairs
whose elements can be any two real numbers
the set "R-three" consists of ordered triples whose elements can be any three real numbers.
And just as the 2-dimensional Cartesian plane was built from two number lines
oriented perpendicular to each other
3-dimensional Cartesian space can be created by taking three number lines
and orienting all three perpendicular to each other.
These three axes are typically labeled x, y, and z
with the positive x-axis pointing out of the page
the positive y-axis pointing to the right
and the positive z-axis pointing up.
Just as in two dimensions, the point where the three axes meet is called the origin.
The origin correspond to the ordered triple (0,0,0).
The three axes can be oriented in any way
as long as all three axes are perpendicular to each other
and they conform the what is called the "right hand rule".
The right hand rule states that if you align the thumb of your right hand
pointing in the positive direction of the x-axis
and your index finger
pointing in the positive direction of the y-axis
then your middle finger should point in the positive direction of the z-axis.
For example, we could arrange the axes so that the positive x-axis points to the right
and positive y points up.
In that case, the right hand rule tells us that positive z must point out of the page.
Or we could orient the axes with positive x pointing up
positive y pointing out of the page
and positive z pointing to the right.
However, the typical orientation is with positive x pointing out
positive y pointing to the right
and positive z pointing up.
In three dimensions, just as in two dimensions
the infinite plane containing the x and y axes is called the xy-plane.
Likewise, the y and z axes create the yz-plane
and the x and z axes create the xz-plane.
These three planes divide the Cartesian space into eight regions called "octants".
The octants are numbered one through eight and are typically labeled using Roman numerals.
Just as the 2-dimensional Cartesian coordinate system
allows us to graphically display ordered pairs of real numbers
as points on the Cartesian plane
3-dimensional Cartesian coordinates allow us to display ordered triples
as points in Cartesian space.
The elements of an ordered triple corresponding to a point are the "coordinates" of that point.
To locate a point in this 3-dimensional space
starting from the origin we move along the x-axis
a distance and direction specified by the x-coordinate.
Then from that point we move parallel to the y-axis
a distance and direction specified by the y-coordinate
and finally we move parallel to the z-axis
a distance and direction specified by the z-coordinate.
This determines the position of our point in Cartesian space.
Another way to look at this
is that the x and y coordinates locate a position on the xy-plane just as in two dimensions
and the z-coordinate specifies the point's distance
above or below that position on the plane.
Now that we have a way to visualize ordered pairs of real numbers as points on a plane
or ordered triples as points in space
the next step will be to create sets of ordered pairs
which represent relations between different types of quantities.
If the quantities can be represented by real numbers
then these relationships can be visualized in 2 or 3-dimensional space.
We will do this in the next lecture
by introducing the concept of a "binary relation".