Three-dimensional coordinate systems level 4
many the formulas established for a two-dimensional coordinate system
can be extended to three dimensions, it turns out
that many of the formulas that you're used to working in R-squared
have natural extensions in R-cubed, the first of these formulas
is the all familiar midpoint formula recall from your previous math classes
that the midpoint of a line segment denoted as P sub M
where M stands for midpoint, joining the points
P-one with coordinates x-one and y-one
and P-two with coordinates x-two and y-two
has midpoint coordinates x sub m and y sub m
equal to the coordinates x-one plus x-two divided by two
and y-one plus y-two divided by two
respectively. In the same manner
the midpoint of a line segment P sub m in space
joined by the points P-one with coordinates x-one
y-one and z-one and P-two
with coordinates x-two, y-two, and z-two
has coordinates x sub m, y sub m
and z sub m equal to x sub one
plus x sub two divided by two, y sub one
plus y sub two divided by two and z sub one
plus z sub 2 divided by two respectively. Notice how we included
a new variable, in this case we included the variable z
for the most part many of the equations that you were introduced in your previous
math classes
can be extended to three dimensions, by just including this additional variable
z
let's go over an example. Find the midpoint
of the line segment connecting the points (4,3,1)
and (-2,5,7). Let's go ahead and plot these points in a
three-dimensional Cartesian coordinate system
just so we can get a visual representation of where these points are
located
in space. The first point has coordinates (4,3,1)
so we plot this point by starting at the origin and moving four units towards the
positive
x-direction, then we move three units towards the positive
y-direction, and lastly move one unit towards the positive z-direction
in the same manner the second point is plotted by starting at the origin
and moving two units toward the negative x-direction then moving five units towards
the positive y-direction
and lastly seven units towards the positive z-direction
this problem is asking us to find the coordinates
of the midpoint, a line segment connecting these two points
so if you imagine a line connecting these two points
in space, we're trying to find the point that is exactly halfway between these
two endpoints
hence the midpoint, also keep in mind
that geometrically the midpoint divides a line
into two equal parts, finding the midpoint is easy
but really understanding what a midpoint represents from a geometric
point of view
is much more powerful and useful when it comes down to it
the coordinates of the midpoint of a line segment is nothing more than the
averages
or the arithmetic mean of the coordinates of the endpoints
so the average of the x-coordinates is equal to four minus two
divided by two which is equal to one. The average of the y-coordinates
is equal to five plus three divided by two which is equal to four
and the average of the z-coordinates is equal to seven plus one
divided by two which is equal to four. So the coordinates of
the midpoint is equal to (1,4,4). Alright
let's move along to the next formula recall that in order to find a distance
between two points P-one with coordinates x-one
and y-one and P-two with coordinates x-two and y-two
in other words the line segment from P-one to P-two
in two dimensions we use the distance formula which was derived by using
the Pythagorean theorem
in the same manner the formula to find the distance
of a line segment in space can be derived by using the Pythagorean theorem
twice, let's go over this simple exercise so we can work out or spatial
visualization skills
say we have two random points in space let's call them point one
with coordinates x-one, y-one and z-one
and point two with coordinates x-two, y-two
and z-two, in addition lets create a box
that includes points P-one and P-two as opposite vertices
where point one is located in the bottom corner of the box
and point two is located in the opposite top corner of the box
this way both points have completely different x, y and z-coordinates
since these specific vertices don't share a common face on the box
if they did then we would be working on a boring two-dimensional problem
if both points where on the same face, okay
in order to find the distance between point one and point two
which is represented by the red line segment we need to relate
the length of this segment by using the coordinates of point one
and point two, notice that we can actually find the coordinates
of the other vertices by using the coordinates of P-one
and P-two, for example I'm going to label the vertex
directly across P-one towards the positive x-axis
as vertex A and label the vertex below point two
as vertex B, now
in order to assign coordinates to these vertices in terms of the coordinates of P-one
and P-two
we notice the following, the front face of the box contains point two
this phase is parallel to the yz-plane this means that every single vertex
in this plane has x-two as their x-coordinate
including vertex A and B, similarly
the faces on the sides of the box are parallel to the xz-plane
and contains point one in the left face and point two
in the right face, this means that every single vertex
on the left face will have the same y-coordinate as point one
in this case y sub one, so vertex A will have y sub one
as it's y-coordinate and every single vertex in the right face
we have the same y-coordinate as point two
in this case y sub two, so vertex B will have y sub two
as it's y-coordinate, and lastly the plane that is located in the bottom
of the box is parallel to the xy-plane this means that every vertex
in this plane has the same z-coordinate as point one
in this case both vertex A and B will have z sub one
as their z-coordinate. Now, in order to find the distance between point one and point two
represented by the red line I need to somehow relate this distance by using
the coordinates of point one
and point two, it turns out that we can relate them by using right triangles
notice that we can form a couple of right triangles in this box
the first right triangle contains the line segment formed by vertex B
and point two, and the line segment formed by vertex B
and point one, these two segments represent the legs
of this right triangle with the right angle being formed with the bottom plane of the
box
and the line segment formed by point B and point two
in addition, the line segment formed by point one and point two
represents the hypotenuse of the right triangle
in order to find the distance we need to know the length of each leg
we can easily find the length of the leg formed by point two
and vertex B this is just going to be equal to the absolute difference between
the points
z-coordinates remember we need the length to be a positive value
so we take the absolute value of the difference of their coordinates
the distance of the other leg is not as straightforward and is a little bit more
involved
I'm going to color this line segment with the color blue
it turns out that this line segment is also the hypotenuse
of another right triangle this right triangle
has as its legs the line segment formed by point one
and vertex A, and the line segment formed by vertex
A and vertex B with the right angle being formed by these two legs
in order to find the length of the blue line segment
which represents the hypotenuse of the second right triangle
we need to find the length of each of its legs once again
we can easily find these, the length of the segment formed by point one
and vertex A will be equal to the absolute difference between the points
x-coordinates and the length of the segment formed by vertex A
and vertex B will be equal to the absolute difference of the points
y-coordinates
so now we essentially have to apply the Pythagorean
theorem to find the length of the blue line
the relations are as follows, the length of the blue line segment squared
has to equal to the length of the line segment between point one
and A squared, plus the length of the line segment between point A and B
squared. We know the lengths of the line segment between point one
and A, it is equal to the absolute difference of their x-coordinates
and we also know the length of the line segment between point A and B
it is equal to the absolute difference between their y-coordinates
simplifying the expression we have the following
now, lets apply the Pythagorean theorem to the other right triangle
that has as its hypotenuse the red line which is
ultimately what we're trying to solve for
so we have that the length of the red line segment squared
has to equal to the length of the line segment between point one
and point B squared, plus the length of the line segment between point two
and B squared. From the first application of the Pythagorean theorem
we found the line segment between point one and B
this was equal to the following expression,
and the length between point two and point B is equal to the absolute difference
of their z-coordinates combining them together
and simplifying we obtain the following expression,
and finally its just a matter of taking the square root of both sides
doing that we obtain the distance formula in three dimensions
between the points P one with coordinates x-one, y-one
and z-one and point two with coordinates x-two
y-two and z-two, notice that the equation resembles that of its
two-dimensional counterpart
the only difference is that we included an extra variable
in this case z, because of the fact that we are dealing with a three dimensional
coordinate system
alright, in our next video we will go over examples
where we make use of this new formula and we will also derive the equation
of a sphere.