Three dimensional coordinate systems. Level two. In the previous video we reviewed the basics
of one dimensional, two dimensional, and three dimensional coordinate systems. In this video
we will begin representing basic equations in 3-Dimensional space.
Before we dive in with equations we need to be able to draw a 3 dimensional coordinate
system and be able to identify and navigate though all of its parts without getting lost.
Lets first start with the traditional 2-Dimensional Cartesian coordinate system. Recall that this
coordinate system is formed by two axis the x axis and the y axis which are perpendicular
to each other. In addition, this coordinate system is divided in to 4 quadrants, the first
quadrant is located here then we move counter clock wise to label the second, third and
4th quadrant. Recall that each quadrant determines the signs on the x and y coordinate of a point.
You can view the previous video with a more in depth look on two dimensional coordinate
systems. For now I want you keep in mind that a 2 dimensional coordinate system represents
a plane. Now when we draw a 3 dimensional coordinate system we usually draw the standard placement
of the axis meaning we only show the positive axis for each variable and we extend the lines
for each axis when needed. So we first draw the positive z and y axis as follows then we extend
the x axis from the common point of intersection of the z and y axis as follows. This is the
standard position of a 3-Dimensional coordinate system keep in mind that we will tend to extend
the lines for each axis when needed. Also keep in mind that all the physical drawings you
make are done in a 2 dimensional sheet of paper this means that you will have to use
your spatial visualization skills to see most of these 3 dimensional graphs. But in these
videos I will create actual 3 dimensional images so you can start getting an intuition
of how these graphs are truly represented. Let’s take a look at our two dimensional coordinate
system, it turns out that this coordinate system represents the xy coordinate plane
in a 3-dimensional coordinate system as shown on the right if you view the plane from above
it’s actually turned a little over 90 degrees clock wise. This means that the first quadrant
is located here, the second quadrant here, the third quadrant in the back and the 4th
quadrant is located here. You really need to wrap your head around this and be able
to view the xy plane in a 3 dimensional system. This way you can translate all of your understanding
of a 2 dimensional coordinates system into the xy plane of a 3-dimensional coordinate
system. In addition keep in mind that there are 2 other planes the yz plane and xz plane.
If you understand how graphs behave in the xy plane you will easily understand how graphs
behave in the yz and xz plane. Its pretty much the same concept Alright. Now lets actually start graphing equations. Recall from your studies of algebra that a
graph in two-dimensional analytical geometry, in other words in the Cartesian coordinate
plane, involving the variables x and y is represented by a curve in R squared. For example the
equation y=3 represents the following set, which is the set of all points (also known as ordered
pairs) in R squared whose y-coordinate is 3. This means that this equation contains all the points whose coordinate is (x,3) the way we represent this is by plotting all the points in the xy plane or in this
case R squared whose y coordinate is 3, doing that we would obtain the following points notice that
every single point has 3 as its y coordinate, now this set of ordered pairs is in theory
infinite so instead of plotting all the possible points we usually represent all of these infinite
points by connecting them with a curve as follows in this case we have a horizontal
straight curve connecting all the points that are generated by the given equation. In addition
this line is also parallel to the x axis and perpendicular to the y axis. In the same manner
the equation x=3 represents the folloiwng set which is the set of all points whose x-coordinate
is 3. This means that this equation contains all the points whose coordinate is (3,y) the way we represent
this is by plotting all the points in the xy plane whose x coordinate is 3, doing that
we would obtain the following points notice that every single point has 3 as its x coordinate,
once again we usually represent all of these infinite points by connecting them with a
curve as follows in this case we have a vertical straight curve connecting all the points that
are generated by the given equation. In addition this line is also parallel to the y axis and
perpendicular to the x axis. Notice that when you have an equation with a single variable
such as y=3 or x=3, the second variable is free to take any value since there were
no restrictions on that variable, this idea will be is very important when we start graphing equations
in a 3 dimensional coordinate system. So let’s go ahead and graph the equation y
=3 in R cubed. Notice that from this point on its key to specify what coordinate system are
we referring to. Is it R squared or R cubed since this will determine how many variables we are going
to deal with and ultimately determine the final graph of the equation. In this case
we are graphing y=3 in R cubed this equation represents the following set, which is the set of all points (also
known as ordered triples) in R cubed whose y-coordinate is 3. This means that this equation contains
all points whose coordinats is (x,3,z), so we first draw the curve for the equation y=3 in the xy plane
as follows, this curve is analogous to the curve in R squared but now we are graphing it in R cubed this line represents all the points whose coordinate is (x,3,0) but remember that the equation y=3 does not
contain any of the other 2 variables x or z this means that these variables are free
to take any values, as a consequence the following points are also part of the set of the equation
y=3 in R cubed these points represent the various values which include the z coordinate. So
all of these points are also points that are generated by the equation y=3. Once again
in theory we have an infinite number of points so we usually connect all of these points
with a surface in this case the equation is represented by a plane. Its essentially a
sheet of paper that is parallel to the xz plane, and is also perpendicular to the y
axis. In the same way, the equation x=3 is represented in R cubed as follows, its also a plane
that is parallel to the yz plane and is perpendicular to the x axis. Finally the equation z=3
will be represented as follows. it’s also a plane that is parallel to the xy plane and
perpendicular to the z axis. At this point we have equations to describe
the coordinate planes of a three dimensional coordinate system, the xy plane can be represented
by the equation z=0, the xz plane is represented by the equation y=0, and the yz plane is
represented by the equation x=0. Alright. Another concept that requires an understanding
of these coordinate planes is the idea of a projection of an object on one of these
planes. Let’s keep it simple and first talk about
the projection of a point on the coordinate planes, say we have a random point in space
let’s call it point P whose coordinates is equal to (x,y,z) if I were to view this point from above and shine a light
directly above this point it will create a shadow that is located in the xy plane directly
below point P, the location of this shadow in the xy plane is called the projection of
P on the xy plane. This point has the same x and y coordinates as point P but has a z
coordinate of 0, remember the equation of the xy plane is represented by the equation
z=0 so any point on that plane will have a z coordinate of zero. So let’s denote the
projection of P on the xy plane as point Q it will have coordinates equal to (x,y,0).
Now if I were to look at point P from the first octant facing towards the yz plane and
I shine a light at point p it will cast a shadow directly across point P, this is the
projection of P on the yz plane and has the same y and z coordinates as point
P, but has an x coordinate of 0, the equation of the yz plane is represented by the equation
x=0 so every point on this plane has zero as its x coordinate lets denote the projection
of P on the yz plane as point R and it will have coordinates equal to (0,y,z).
Lastly, if I were to look at point P from the first octant facing towards the xz plane
and I shine a light at point P it will cast a shadow directly across point P, this is
the projection of point P on the xz plane and has the same x and z coordinates as point
P, but has a y coordinate of 0, the equation of the xz plane is represented by the equation
y=0 so every point on this plane has zero as its y coordinate lets denote the projection
of P on the xz plane as point S and it will have coordinates equal to (x,0,z).
In a later topic this concept of a projection is going to be useful as we tackle more challenging
concepts in multivariable calculus. Alright in the next video we will continue representing
basic equations in 3 dimensional space.