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Calculus III: Three Dimensional Coordinate Systems (Level 3 of 10) | Planes, Cylinder

July 31, 2019


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

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34 Comments

  • Reply Math Fortress August 28, 2012 at 2:32 pm

    No Problem. Let every one know about mathfortress.

  • Reply Ahmed Medhat November 16, 2012 at 11:59 am

    cool, so cool,
    thanks.
    in the last example the,unintentional, edge makes the graph a bit ugly.

  • Reply Math Fortress November 16, 2012 at 4:37 pm

    Are you referring to the bold Coordinate Axis? I noticed that it became bold after the animation.

  • Reply Susan Burgess January 20, 2013 at 4:13 am

    Thanks for your help with 3-D vectors…I couldn't find as much information anywhere else

  • Reply Math Fortress January 20, 2013 at 4:44 am

    No problem glad that the video was useful. Let every one know about math fortress!

  • Reply Abdul Moaeed Raja April 4, 2013 at 11:54 pm

    great video man! I finally understand how to graph in 3-D! 😀

  • Reply randomtryouts April 15, 2013 at 8:02 pm

    Thanks, exactly what I needed; finally makes sense. Now I can continue studying in peace!

  • Reply Math Fortress April 15, 2013 at 8:09 pm

    Glad the video helped you out. Let every one know about Math Fortress!

  • Reply Math Fortress April 15, 2013 at 8:10 pm

    😀

  • Reply Harish Babu Kankanala July 3, 2013 at 4:30 pm

    great video…..
    I have learned many new things from your videos…..thank u.

  • Reply Math Fortress July 3, 2013 at 10:45 pm

    No problem! Let every know about Math Fortress!

  • Reply aisha madondo July 16, 2013 at 9:55 pm

    great videos….

  • Reply Math Fortress July 17, 2013 at 1:27 am

    Thanks! Spread the word about Math Fortress!

  • Reply David Dolynny May 7, 2014 at 10:02 pm

    you are very good at explaining! I have only watched 4 videos so far, they have been excellent, good job not being annoying like other math videos

  • Reply Maykon Beraldi November 6, 2015 at 1:05 pm

    Thank you for helping. You explained it very well.

  • Reply Spider Man November 21, 2015 at 12:56 pm

    I am going to make sure to share all of your stuff and spread them as you are life savior to me , and i can now live in peace with my math, one thing i will ask though do you have a lesson on function approximation using taylor series or maclaurin and what is the best way to understand that topics what do u recommend

  • Reply Tulasidhar Vemuri December 7, 2015 at 3:47 pm

    show that the points (-2,3,5) and (1,2,3) (7,0,-1) are collinear by using slopes

  • Reply Bad Hombre December 28, 2015 at 7:30 am

    You are a master, my man!

  • Reply 季明強 February 13, 2016 at 8:57 am

    Very good. Do you have the video for the complex variable ?

  • Reply Rene Sebastian February 17, 2016 at 7:31 pm

    This video helped me a lot, especially with visualizing this. Could you maybe do one on the regions bound by different function in R^3 (i.e from R^2 to R^3)? That stuff is hard to picture! Definitely spreading the word on this!

  • Reply Yusuf Paiman May 17, 2016 at 6:55 am

    What app do you use for giving motion to these graphs??? (Please tell me the name)

  • Reply Fonseca September 1, 2016 at 7:44 pm

    Thank you for helping me understand !

  • Reply Antonio Molina September 6, 2016 at 4:21 am

    Thank you so much for this my teacher is great but honestly I was lost until I saw your videos. I'm at # 3 so far and will watch them all!!! It really is helping me out I really appreciate it. I will spread the word to my classmates!

  • Reply Ali Abu September 15, 2016 at 1:59 pm

    Whats app do you use to make those graph?

  • Reply Fox Back November 11, 2016 at 12:16 pm

    Thank you. Best regards from India!

  • Reply Rajan Bawa February 27, 2017 at 6:25 am

    great

  • Reply Abbas Mazari April 3, 2017 at 7:56 pm

    The nicest way to explain 3-DCS .
    I have been inspired to watch these lecture videos and currently solve my exercise.
    wonderful….!

  • Reply Tinh Apple September 12, 2017 at 3:41 am

    WOW. Amazing!!!!!!!!!!

  • Reply fatin syafiqah October 1, 2017 at 1:17 pm

    I want to make education video like this. what software did you use?

  • Reply NOor Idris October 18, 2017 at 5:34 pm

    amazing thank you so much

  • Reply Jn Githinji October 28, 2017 at 1:12 am

    Beautifully explained. Thank you!

  • Reply dave stevens December 23, 2017 at 10:19 pm

    I have to say… As well as everyone else…. Great Job!!!
    I have been out of the education ( graduated school) for many years, I recently found myself interested in becoming refreshed with your videos… Who says you can never teach a old dog new tricks… May not be new tricks.. But the process you have demonstrated for others is out of this world… Good Job… My daughter will be excited to know she can learn from you and to excel in higher education… Math to. Me has always been exciting and fun… You just proved me right… I am extactic,
    Thank you again

  • Reply Math Fortress June 27, 2018 at 3:13 am

    0:19 Equations in R^3

  • Reply Peter Tailor November 17, 2018 at 2:49 pm

    Thank you!
    How do you tell which one is the independent variable? (3rd exmpl)

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