Three dimensional coordinate systems level three.
in the previous video we started to describe basic equations
in R cubed. Specifically describing equations on a three-dimensional Cartesian
coordinate system
in this video we will continue describing basic equations
in three-dimensional space. Let's start by graphing the equation y = x
in R-squared. Since this graph will provide us with a starting point
to successfully graph it in R-cubed. For starters,
this equation is represented by a line in R squared
and has the form y = mx + b
where the slope is equal to one and the y-intercept
is equal to zero. So we typically graph these equations by first plotting the
y-intercept
in this case it's equal to zero, then we use the slope
which in this case is equal to one, to generate a second point.
recall that a slope of positive one means that we move in the positive
y direction
one unit and move in the positive x-direction
one unit. Remember the slope can be interpreted
as the rise over the run. Once we have our two points we go ahead and connect them
obtaining the following straight curved. Now
if we were to graph this equation in R-cubed we would obtain the same curve in the
xy-plane
this curve has a y-intercept of zero and a slope of
positive one, but notice that the variable z is not mentioned in the
equation
meaning we have not specified z in any way
so we must assume that z can take on any value
this means that any particular value of z will get a copy of this line
both the positive values of z such as z = 1
z = 2, z = 3, and so on and negative values of z
such as z = -1, z = -2,
and z = -3 and so on. In this sense the graph represents a vertical
plane
that lies over the line given by the equation y = x
in the xy-plane. Notice that we started by drawing the curve
on the plane that contained the variables for which the equation was defined
in this case the equation was defined but the variables x and y
so we first graphed it on the XY plane, then we drew copies of this curve
on both the positive and negative direction of the missing variable
which was not expressed in the equation in this case the variable that was not expressed
was z
remember variable that are not explicitly expressed in the equation
are assumed to attain any value unless mentioned otherwise.
we will go over equations in which the variables are restricted
in a later video. Alright, lets spice things up a bit
and graph the equation z = -2y + 1.
once again, this equation has the form
y = mx + b, but instead we have the equation
z = my + b, where z is the dependent variable
and y is the independent variable, we essentially treat the equation
just like we did in the previous example, but instead of graphing it
in the xy-plane we are actually going to graph the curve on the yz-plane
as follows, here we have a two-dimensional representation
of the yz-plane we're going to graph the curve in this plane
and then translate it into R-cubed, so let's go ahead and plot
what used to be the y-intercept which in this case is now the z-intercept on the
yz-plane
so we plot the first points at z = 1, then we use the slope,
in this case its negative two, which can be interpreted
as negative two over one or in this case the rise over run
and use it to generate a second point, so from the z-intercept
we go ahead and move two units down along the negative z-axis
and move one unit to the right along the positive y-axis
then we connect the points with a curve as follows
alright, having graphed the equation in the yz-plane
we're ready to translate it into the yz-plane of a three-dimensional
Cartesian coordinate system
its the same idea, but now we need to be able to navigate
along the yz-plane of this three-dimensional coordinate system
the yz-plane is located here, the positive y-axis is located here
and the positive z-axis is here, so we literally draw the same curve
we just graphed in R-squared onto the yz-plane
of R-cubed as follows
next we notice the variable x is not expressed in the equation
and there are also no restrictions this means that x can take on any value
as a consequence, at any particular value of x
there will be a copy of this line both on the positive values of x such as
x = 1, x = 2, and x = 3,
and so on, and the negative values of x such as
x = -1, x = -2, x = -3,
and so on, in this sense the graph represents a plane
that intercepts the z-axis at z = 1
and has a negative slope the plane is inclined
with a slope of negative two over one
alright, lets try the next example. This one requires graphing an equation
on the xz-plane. Let's graph the equation
2z + 1 = 3x, notice that this equation is not in slope
intercept form so we first need to arrange the equation into the slope intercept
form
of a line, notice that all the concepts you learned in your previous math classes
come back in some shape way or form
this will be true as we dive deeper into multivariable calculus
okay, in order to write the equation in slope intercept form
we first need to solve for the dependent variable in this case the dependent
variable
is z and the independent variable is x
this is similar to the way we solved for y when dealing with an equation
in the xy-plane back in your pre-calculus class
so we go ahead and subtract one from both sides
then we divide both sides by two, doing that
we obtain the equation z equals three half x minus one half.
let's go ahead and graph this equation directly into the three-dimensional coordinate
system
you first need to graph the curve on the xz-plane so looking at the equation
we see that the z-intercept is negative one half
so we first plot this point as follows
then we noticed that the slope is positive three halves
so we use this slope to generate an additional point, so starting from the
z-intercept
we move three units towards the positive z-axis and two units towards the positive
x-axis, then we go ahead and connect a curve connecting both points
as follows. Having created our first curve
we go ahead and make copies of this curve at positive values of y
and at negative values of y, remember when a variable is not expressed
in an equation and no restrictions are mentioned the variable is free to take
on any value
lastly, we connect all the curves together and obtain the following plane
that has a z-intercept of negative one half and is also inclined
with a slope of positive three halves
alright, these three examples were designed to illustrate the fact that you
can use all your knowledge and experience from graphing equations
in a two-dimensional coordinate system and translate them into three dimensions
it might take you some time to get used to but like in all your math classes practice is key
in developing this skill. In a later video we will actually develop a more powerful way
of describing planes
in 3D space. For now, these examples were designed so you can start obtaining an idea
of how to navigate through this three-dimensional coordinate system
alright, lets try graphing something other than planes
let's give it a shot and graph the following equation.
x squared plus y squared equals four. Notice that the variables
that are explicitly expressed are the variables x and y.
so we are going to first graph this equation in R squared
specifically on the xy-plane. Recall from your studies of precalculus that this
equation represents a circle
centered at the origin with radius two, so its graph is represented in R-squared
as follows. Now if we want to graph this equation in R-cubed
we go ahead and graph this curve on the xy-plane as follows.
next we notice that the variable z is not expressed in the equation
remember this means that every single value of z both positive and negative
will have copies of this curve which has the shape of a circle
finally we connect the curves with rulings as follows
this equation represents a cylinder centered in the z-axis
so in R-squared the equation represents a circle
and in R-cubed it represents a cylinder once again
this example illustrates the importance of specifying what coordinate system
we are referring to, since it ultimately determines the graph the equation
aright, let's end the video by graphing one final equation
lets graph the equation z equals x squared
notice that the variables z and x are expressed in this equation
so we're going to go ahead and graph them on the xz-plane
in addition, notice that this equation looks like the equation of a parabola
of the form y equals x squared but in this case
the dependent variable is z and the independent variable is x
other than the variable z, this equation pretty much represents a parabola
in the xz-plane and is graphed as follows
this is how this equation will look like in R-squared
next we go ahead and graph the same curve on the xz-plane
of R-cubed as follows
notice that the variable that is not expressed is y
this means that every single value of y will have a copy of this curve
since its free to take any value of y, in addition
no restrictions have been imposed on the variable y. Next we go ahead
and connect the curves as follows, doing that we obtain the following surface
this surface is called a parabolic cylinder
we will study this surface and many more in a later video
for now, keep in mind that equations in R-squared are usually
represented by curves on a plane and equations in R-cubed
are usually represented by a surface but not always
they can also be represented by curves in space
as we will see in a later video. Alright
in our next video we will start exploring common formulas associated
with a three-dimensional coordinate system