Three-Dimensional Coordinate systems (Level 10)
In this final video on three dimensional coordinate systems, we will go ahead and learn how to
graph equations and inequalities that are restricted to a given interval. Recall from
your prior math classes that the graph of a function can be restricted to certain values
of its domain and range by specifying the restriction using inequalities. For example
if we were the graph the equation y equals x squared in R-squared it will look like
this, it's essentially a parabola that attains a minimum value at x = 0. Now, say that I only
want to graph this equation starting at x = 1 to positive infinity, as follows
the way we mathematically express this graph is by denoting an inequality that describes this
restriction, in this case we want to graph the equation y equals x squared when x is
greater than or equal to 1, if we don't want to include the value at x = 1 then we write
x > 1, then the closed circle is changed to an open circle to let us know that we do not
include the value at x = 1. In the same manner we can restrict the graph of an equation in
three dimensional space. Let's go over the first example.
Example 1: Describe the region of R-cubed represented by the equation.
y equals 0 when z is greater than or equal to 0.
Recall from the previous videos that the equation y = 0 represents the xz-coordinate plane and
is graphically represented as follows, next we go ahead and apply the restriction in this
case we only want that part of the graph where the values of z are greater than or equal to zero,
this means that we only want that part of the graph that is located on the regions where
the value of z are positive in this case everything above the xy-plane. So our graph
will look like this, notice that there was no restrictions on the values of x remember
this means that the values of x are free to attain any value. If on the other hand we
add an additional restriction like x is greater than or equal to 0, then we would need to
erase that part of the graph y = 0 where the values of x are negative as follows, because
we only want that portion of the graph where the values of z and x are positive. In this
is the region that is graphed it's essentially the first quadrant of the xz-coordinate plane
if we were to look at it from this direction Alright, let's try the next example:
Example 2: Describe the region of R-cubed represented by the equation.
x squared plus y squared equals 4, when z is between negative 2 and 2, inclusive.
When dealing with these type of problem it's always a good idea to ignore the restrictions
for now and graph the equation as if there was no restrictions. Recall that this equation
represents a cylinder in R-cubed, the variables expressed in the equation are x and y so the
expression x squared plus y squared equals 4 represents a circle in the xy-plane, in
addition, since we are graphing this equation without any restrictions we create a copy
of this circle on every single value of z as follows.
So technically this cylinder extends all the way to positive infinity in the positive z
direction and negative infinity in the negative z direction. Next lets go ahead and apply
the restrictions, notice that the inequality describes an interval in this case the acceptable
values of z that this equation can attain are all the values between negative 2 and
positive 2 inclusive, This means that our cylinder will only have copies of the circular
trace between these values of z as follows, so we now have a cylinder that has a height
of 4 units. What if we wanted to graph only the left half of the cylinder? What restriction
would I have to add? In this case we only want that part of the graph where the values of
y are negative so we need to add the following inequality y is less than or equal to 0. If
I want the right half then I include the inequality y is greater than or equal to zero, notice
that in both cases we are including 0 if you don't want to include zero then we remove
the equal sign and add dashed lines on the sides of the cylinder as follows.
As you just saw we can control what portion of the graph we want to focus on by restricting
certain values with inequalities.
Alright, the next types of graphs we will focus on are the graphs of inequalities themselves.
Recall from your previous math classes that when you graph an inequality in R-squared
we are trying to find a specific region whose points satisfy the inequality. For example
say we want to graph the inequality y is greater than or equal to x squared, the first step
is to graph the inequality as if it were a regular equation, as follows, next we need
to determine the region above or below the graph that satisfies the inequality, in this
case we want to shade the region were the values of y are greater than the values of
x squared, this means that we need to shade the region above the graph as follows. Recall that this
region determines a set of points that satisfies the inequality so if you were to choose any
point on this region it should satisfy the inequality in other words you should obtain
a true statement when you substitute the coordinates of any point located in this region into the
expression. Notice that the set of points that will make this inequality a true statement
also includes the points located on the parabola itself, if we were to change the inequality
to y > x squared then we would graph the function using dashed lines this means that we don't
include the points located on this function itself. Now let's go ahead and graph some inequalities
in R-cubed. Example 3: Describe the region of R-cubed
represented by the inequality.
x is greater than 3
Alright, the first step is to graph the inequality as if it were an equation so we are essentially
graphing the equation x = 3, recall that this equation represents a plane that is parallel
to the yz-coordinate plane and located 3 units in the positive x direction, Next we go ahead
and apply the inequality, in this case the values of the x have to be greater than 3,
this region consist of all the points in front of the plane x = 3 not including those points
on the plane. Alright let's try the next example. Example 4: Describe the region of R-cubed
represented by the inequality.
Here we have an inequality that represents an interval. The way we deal with these types
of inequalities is by graphing the function in the middle of the inequality by equating
it with the left value and the right value, so we first want to graph z = 0 and z = 6,
These equations represent planes that are parallel to the xy-plane.
So we go ahead and graph them.
Next we apply the
inequality, so the region represented by the inequality are all the points between these
two planes including the points on the bottom plane but not including the points located
on the top plane. Alright, let's try the next example:
Example 5: Describe the region of R-cubed represented by the inequality.
x squared plus z squared is less than or equal to 9
As always let's go ahead and graph the inequality as if it were an equation, this equation represents
a cylinder with circular traces of radius 3 located in the xz-plane, and centered in
the y-axis. Next, lets apply the inequality, we want to shade that part of R-cubed where
x squared + z squared is less than or equal to 9, another way of representing this inequality is by
taking the square root of both sides as follows, so we essentially want the set of all points
whose distance from the y-axis is at most 3, these points are located inside of the
cylinder we also want to include those points that are located on the cylinder.
Alright, lets try the next example: Example 6: Describe the region of R-cubed
represented by the inequality.
Alright, once gain we have an inequality defined by a given region, in this case the middle
function represents a sphere and the numerical values represent the smallest and largest
value of the radius that the sphere can attain. So let's go ahead and graph the equation x
squared plus y squared plus z squared equals 1 and the equation x squared plus y squared
plus z squared equals 4 we essentially have two concentric spheres centered on the origin.
Next, let's go ahead and apply the inequality, lets rewrite the inequality by taking square
root though out as follows. So we want the set of all points that are at least
1 and at most 2 from the origin, in other words we want to shade the region that is
between these two spheres. Including the points that are located on the surface of the sphere
with a radius of 2 and the sphere with a radius of 1.
Alright, say that we only want the upper half of these spheres in other words all the points
that are above the xy-plane how can we accomplish this? We can denote these points by including
an additional inequality in this case we want all the points that are above the xy-plane
or all the points that contain a positive z-coordinate, so we go ahead and include the
inequality z is greater than or equal to 0, so we can restricted the values of z to take
on positive values only. Doing that we would obtain the following graph.
Alright, Let's go over the final example. Example 7: Write an inequality to describe
the region: The region inside a cube centered in the origin
with side length equal to 4. In this problem we are asked to come
up with the inequalities that describe the given region. We basically want the set of
points that are located inside a cube of side length equal to 4. We can start by generating
each of the faces of the cube since each face represents a plane. The top and bottom face
can be obtained by graphing the equation z = 2 and z = -2, as follows, in addition since
we want all the points that are between these two planes we want to set up an inequality
that represents the region between these two planes we express this region as follows,
next we want an expression that defines the sides of the cube, the sides that are parallel
to the xz-plane are represented by the equation y = -2 and y = 2 respectively and the inequality
that defines the region between the planes is expressed as follows, and finally the front
and back faces of the cube are parallel to the yz-plane and can be obtained by graphing
the equation x = 2 and x = -2 respectively and the region between these two planes can
be expressed as follows, so the inequalities that define the set of all points inside a
cube centered on the origin with side length equal to 4 are defined by these three inequalities.
Alright, now that we have laid the foundations in navigating and maneuvering a three dimensional
coordinate system we are ready to learn about vectors which will provide us with the tools
necessary to tackle on the rest of calculus 3.