This is a long module in which we build up some powerful geometric ideas and techniques that let us solve many different problems. In this module we use these techniques to go back-and-forth between a three dimensional world and two dimensional pictures. The links below enable you to go directly to different parts of this module.
One way to build up a mental picture of a three dimensional object is to walk around (or fly around) the object looking at it from different points-of-view. You can see different things from different vantage points and by putting together information from different vantage points you can synthesize a three dimensional picture.
The view that you get of an object depends on the direction from which you view it and it also depends on your distance. The two photographs below show our familiar Tinkertoy house viewed from two different distances from approximately the same direction. Which photograph was taken from close by and which photograph from far away? Check your answer by looking at your Tinkertoy house from nearby and far away.
In this module we will develop mathematical techniques for looking at a three dimensional object from a great distance from various different directions. Later we will develop techniques for looking at a three dimensional object from close by. The same tools that we develop for looking at a three dimensional object from different directions will enable us to solve the problem at the end of the preceding module -- How can we determine the point at which two lines based on possibly flawed measurements "intersect" when, because of measurement error, they don't really intersect?
We begin by talking about the length or magnitude of
a vector in R2. The figure below shows a point in
R2
represented by a vector V = (x, y).
The magnitude, ||V||, of this vector is the
distance from the origin to the point -- by the Pythagorean Theorem this is.
The figure on the left below shows a point in R3 represented by a
vector, V = (x, y, z).
The magnitude, ||V||, of this vector is the
distance from the origin to the point and can be computed by using the
Pythagorean Theorem twice. First we look at the right triangle in the
xy-plane whose vertices are at the origin, the point (x, 0, 0),
and the point L = (x, y, 0). This triangle is colored red in the
figure below on the right.
By the Pythagorean Theorem the length of the vector L is
Next we look at the right triangle whose vertices are at the origin, and the
points L and V. This triangle is colored blue in the
figure above on the right. By the Pythagorean Theorem the length of
the vector V is
The distance between two points,
U and V, is the length, ||V - U||, of the movement,
(V - U), required to go from one to the other.
Length or magnitude is an important mathematical
concept that can be used in many different vector spaces. Click on the
button below now for a general mathematical treatment of this topic
as part of the Mathematical Infrastructure. In the rest of this module
we use ideas from this Mathematical Infrastructure section. Because we use
these general results, the ideas we develop here in two and three dimensions
can be applied in more exotic settings.
Given a vector V it is often useful to have a vector U that
points in the same direction as V but has length 1. A vector
of length 1 is called a unit vector.
Theorem:
If V is any vector then
is a unit vector pointing in the same direction as V.
Proof:
Notice that this proof relies only on the magnitude properties
discussed in the Mathematical Infrastructure section on
Length and Magnitude.
The next example is one of the key examples in mathematics, especially if you
are a frog.
Once upon a time a frog
was sitting on a lily pad located at the point represented by the vector
She was very hungry. Fortunately
for the frog an ant
who lived at the origin set out on a walk going in the
direction of the vector
as shown in the movie below.
Since the frog knew some mathematics she knew that as the ant walked along,
a typical point on his path would be of the form
and that the distance from the frog to the ant would be
The frog wanted to eat the ant when he was a close as possible -- that
is, the frog wanted to find the value of t that minimized
the function D(t) or, more simply, the function
So the frog used a little bit of calculus
and decided to eat the ant when he reached the point
But the the ant was quite slow and the frog was very, very hungry, so she
decided to check her work with a little bit of geometry.
The frog noticed that when she ate the ant with her tongue, if the ant was
as close as possible the tongue would make a right angle with the frog's path
in the direction V as shown in the figure below.
The length of the side from the origin to the point where the ant is zapped
by the frog is
and using the fact that the vector
is a unit vector pointing in the same direction as V the frog saw
that she should eat the ant when the ant reached the point.
Putting these two solutions together our talented frog saw that
This led our frog, whose name by the way was dot, to define a new
kind of product, called the frog product, of two vectors by
Mathematicians have since adopted the terminology dot product rather
than frog product to honor the frog's contributions to mathematics.
Using the dot product our frog now knew that she should eat the ant when the
ant reached the point
The dot product is an important mathematical
concept that can be used in many different vector spaces. Click on the
button below now for a general mathematical treatment of this topic
as part of the Mathematical Infrastructure. In the rest of this module
we use ideas from this Mathematical Infrastructure section. Because we use
these general results, the ideas we develop here in two and three dimensions
can be applied in more exotic settings.
The frog and the ant before calculus
The frog and the ant after calculus
Copyright c 1995 by
Frank Wattenberg, Department of Mathematics, Montana State University,
Bozeman, MT 59717