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.