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