Welcome to a lesson on cylindrical coordinates. In this video we'll take a look at
defining cylindrical coordinates and also how to convert between cylindrical and rectangular coordinates.
In a cylindrical coordinate system, a point P in space.
Is represented by the order triple, r, theta and z
R theta is the polar representation of a point in the x, y plane.
Where r is a directed distance from the origin in the x,y plane.
And theta is the angle counterclockwise, from the pole or positive x-axis in the x,y plane.
And z is a directed distance up or down.
From r theta in the x,y plane.
So this might be a little bit small but if we consider the location of point P in space.
R would be the distance from the origin or pole in the x,y plane.
Theta would be the angle of rotation from the positive x-axis or polar axis.
Which would bring us to here.
And then the z coordinate would tell us whether to go up or down.
Let's take a look at an animation that illustrates this a little bit better
As we increase r were moving along with the positive x axis are polar axis.
As we increased theta we rotate counterclockwise.
From the polar axis or positive x-axis as we see here.
And as we increase or decrease
z, we will either move up or down. And here you can see why it's called the cylindrical coordinate system.
Any point could be viewed as on a cylinder and here's a location at that point in space.
Using some trigonometry we can come up with these equations that relate cylindrical and rectangular coordinates.
We have x equal r cosine theta.
Y equals r sine theta, z is equal to z.
And we also have r squared equals x squared plus y squared.
Tangent theta equals y over x and again z equals z.
Let's take a look at doing some of these conversions.
Here we want to convert the point one one three
to cylindrical coordinates. And in this problem they do want us to express the point in two different ways
So were given xyz and we need to determine
r theta and z. Lets start by determining r.
R squared is equal to x squared plus y squared
So here we have one squared plus one square.
So we have r squared equals 2.
Take the square root of both sides
Remember in polar coordinates r can be positive or negative.
But for our first point let's go ahead and use the positive value of r.
So we'll let r equal the square root of two. So for our polar coordinates.
We can use r as the square root of two. Remember z is equal to z. So we know the z coordinate would be three.
And now he determine the value of theta and we can use the second equation here, where tangent theta is equal to y over x.
So tangent theta is going to be equal to one over one.
This should remind us of a forty-five, forty-five, ninety right triangle.
Remember there's an infinite number of angles that have a tangent function value of one.
we'll go ahead and use the most obvious one where theta would have to be forty-five degrees.
Or pi over four radians. So this would be one way to express this point in cylindrical coordinates.
But remember it does ask us to express this using two different ways.
So going back to what we know about polar coordinates.
If we change the sign of r using a negative value of r.
We have to plot an angle that would be different than pi over four.
So if we add a hundred and eighty degrees or pi radians to this.
It would be the same location in the x,y plane.
If we're using a negative value of r remember when r is negative you plot the point in the opposite direction.
So here we could use five pi over four radians.
Instead of pi over four radians and then of course z would be the same value.
So there's two ways to express the point one, one, three.
Using cylindrical coordinates, in this example want to convert,
the cylindrical coordinates of four pi over three negative two to rectangular coordinates.
Here's a location at that point plotted in cylindrical coordinates.
We go out four units from the pole rotate pi over three radians and then down two units.
So for rectangular coordinates we want to determine x,y and z, given r, theta and z.
so the x-coordinate is going to be equal to r times cosine theta.
That will be four times the cosine of pi over three radians.
Just to refresh our memory that would be a sixty degree angle.
So using the thirty, sixty, ninety, right triangle.
Here's our sixty degree angle the cosine of sixty degrees or pi over three radians would be one-half.
So we have four times one-half.
Which would give us two for the x-coordinate.
The y coordinate it's going to be equal to r sin theta. So we have four times the sign of pi over three radians.
So we're gonna have four times. The sine of pi over three would be square root three over two.
Well this simplifies here so we have two square root three.
And that's all we need because remember that z is equal to z.
So the rectangular coordinates for this point.
Would be two, two square root three,
and negative two for the z coordinate.
Now one last thing I want to mention about cylindrical coordinates. We will talk about
cylindrical equations in the next video. But there are some surfaces that are very convenient to represent,
using cylindrical equations. Here we have r equals three,
which creates a circular cylinder. R equals z would be a cone.
R equals the square of z would be a paraboloid.
And theta equals pi over four would be a plane,
as we see here. So we'll talk more about circle equations in another video.
I hope you found this video helpful. Thank you for watching!