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Mathematical Expression Editor
We talk about basic geometry in higher dimensions.
The word geometry can be broken into geo meaning “world” and metry meaning
“measure.” In this section we will tell you what our mathematical “world” is, and how
we “measure” it.
Higher dimensions
In our previous courses, we studied functions where the input was a single real
number and the output was a single real number. Note, the word “real” is being used
in a technical sense:
A real number is a number that has a (possibly infinite) decimal representation.
The set of all real numbers is denoted by .
Which of the following are real numbers?
When we say a function maps a real number to a real number, we write: When
working in two dimensions, we need a way of talking about ordered pairs of numbers.
We denote the set of all ordered pairs of real numbers by . When working in three
dimensions we denote the set of all ordered triples of real numbers by . In
three dimensions we have three coordinates axes, the -axis, -axis, and -axis:
The axes point according to the right-hand-rule:
Of course you will need to “spin” your hand around to align your pointer-finger with
the -axis and your middle-finger with the -axis. Then your thumb will point in the
-direction.
Which of the following axes are aligned according to the right-hand rule?
Point the “pointer finger” of your right hand in the positive direction of
the -axis while simultaneously pointing your “middle finger” in the positive
direction of the -axis. Your thumb will point in the positive direction of the
-axis.
Basic plotting
To plot a point in , you move in the -direction, in the -direction and in the
direction.
Of course, we’re going to be plotting many points. We typically described groups of
points, as those that satisfy a given equation involving , , and . Here is a place
where working in three dimensions is really different from working in two.
In , any equation involving and/or draws a curve.
In , any equation involving , , and/or draws a surface.
The most basic surface in is a plane.
Which equation describes the -plane?
For every point on the -plane, the -coordinate is zero.
Can you describes the solution set of in ?
It’s a horizontal line.It’s a vertical
line.It’s a plane parallel to the -plane.It’s a plane parallel to the -plane.It’s a
plane parallel to the -plane.
consists of all those points where , but and are allowed to be anything.
Another way to think of the point is as the intersection of the planes , ,
.
Move the point around below to see the planes that define it.
While three planes need not intersect at all, the intersection of two (nonparallel)
planes is a line.
What is the intersection of the -plane and the -plane?
The -axis.The -axis.The -axis.
What is the intersection of the -plane and the -plane?
The -axis.The -axis.The -axis.
What is the intersection of the -plane and the -plane?
The -axis.The -axis.The -axis.
Distance and spheres
So the objects in our geometry are made of points, and now we must tell you how we
plan to “measure” objects. To do this, we’ll use our old friend, the distance
formula.
Given two points and in , the distance between them is given by:
This is nothing
more than a corollary of the Pythagorean Theorem. Plot the points and :
We may now construct a right triangle with horizontal side length and vertical
side length whose hypotenuse is the shortest path between the two points:
By the Pythagorean Theorem, the length of this path is given by
What is the distance between the points and in ?
On a completely related note, what’s the most famous theorem in mathematics? I’ll
tell you: The Pythagorean Theorem. In essence, the distance formula is The
Pythagorean Theorem. Let’s see if we can explain why the Pythagorean Theorem is
true.
Given a right triangle,
we have that
Given a right triangle with legs and , and hypotenuse , we can make
the following squares each with side length :
Since both squares above have side length , both large squares have the same area.
Moreover, if we remove the triangles:
Both diagrams must still have the same area, since we removed an equal amount
from each diagram. Hence .
The distance formula also extends to higher dimensions:
Given two points and in , the distance between them is given by:
To see why this is
true, draw a picture!
Ignoring the vertical components of the points, we can make a right-triangle whose
legs have lengths and .
Thus by the Pythagorean Theorem, the length of the hypotenuse of the right-triangle
above is: Now, moving this segment up, we can form another right-triangle where
the legs have lengths and :
The length of the hypotenuse of this new triangle is the distance
between and . Again by the Pythagorean Theorem, we see
Thus the distance formula in is indeed what we claimed.
What is the distance between the points and in ?
In general we can extend this notion of distance to :
Given two points and in , the distance between them is given by:
What is the distance between the points and in ?
Circles and spheres, disks and balls
Let me remind you what the definition of a circle is:
A circle is the set of points in
that are a fixed, nonzero, equal distance from a given point , where is the center of
the circle.
Is the center point of a circle part of the circle?
yesno
Well, the distance from the center of a circle to the center of the circle is zero,
and the radius of a circle is nonzero, so the center is not part of the circle.
From the definition of a circle, we see that it is intimately related to the distance
formula. Indeed, it is also the case the equation of a sphere is related to the distance
formula in :
The equation for a circle of radius centered at the point in is The equation for a
sphere of radius centered at the point in is In general, the equation for a
-dimensional “sphere” of radius centered at the point in is
This follows directly
from the distance formula since a -sphere is the set of points that are equidistant
from a given point in .
In general, a -dimensional “ball” of radius centered at the point is given by the
inequality
Here the “” fills-in the -dimensional sphere.
The equation has a solution set in which forms a sphere. What is the center and
radius of this sphere?
The expression is the square of the distance from to . If the square of the distance
is , then that distance is . Since the solution set of this equation is all points which
are a distance of away from , then this is a sphere of radius centered at
.
Things really get interesting when we have both spheres and planes around. Spheres
can intersect planes at no points (if they are missing the plane), at one point (if they
are “just” touching the plane), or at an infinite number of points (here the
intersection is a circle).
Let’s see some examples.
How many points are on the intersection of and the -plane? Describe the intersection.
The first thing we should note, is that the surface defined by is a sphere, of radius ,
centered at the point . Moreover, the -plane is the plane . So, we have to solve the
following system of equations By plugging into the first equation, we obtain that
So, the sum of the two squares on the left is equal to . This implies that both terms
on the left are equal to . Therefore, the system has a unique solution, namely
.
That wasn’t too bad, let’s see another.
How many points are on the intersection of and the -plane? Describe the intersection.
The first thing we should note is that the surface defined by is a sphere, of radius ,
centered at the point . Moreover, the -plane is the plane . So, all the points on the
intersection satisfy the system of equations By plugging into the first equation, we
obtain that Hence, the intersection consists of infinitely many points and they form
a circle of radius . The circle of intersection is centered at , in the plane
.
For our last example, we’ve left the easiest case of all.
How many points are on the intersection of and the -plane? Describe the
intersection.
The first thing we should note, is that the surface defined by is a
sphere, of radius , centered at the point . Moreover, the -plane is the plane .
Since the plane is units away from the center of the sphere, and the radius
of the sphere is , there are no points on the intersection! We could reach
the same conclusion by setting up a system of equations By plugging
into the first equation, we get that which implies that the system has no
solutions.