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.