You may remember
that in Chapter 2: Satellite Orbits, Section 1: Polar Coordinates, Part 2:
Cartesian to Polar Coordinates we gave a brief description of vectors in
the Cartesian plane before moving on to discuss polar coordinates. If we think
of a point in the plane as the directed "arrow" starting at the origin
and terminating at a point P, thus.
This "arrow" really represents a parallel motion 
of the entire plane that carries the origin to P and that has
this property:
If A
and B are points in the plane, then the segment connecting 
is parallel to the segment connecting 
and these segments have the same length.
When
we think of a point in the plane P as its associated mapping:
we will call it a "vector." Consider two points in the plane P
and Q. They have associated to them parallel translations 
where
|
|
We saw that from properties of parallelograms that the compositions are equal:
|
|
This last fact allows
us to define the sum of two points in the plane (vectors) P and
Q as the vector R, where
|
|
This definition does
not yet depend on the choice of rulers and their coordinates. Pictorially this
definition is easy to visualize in terms of the parallelogram generated by the
vectors:
Now suppose we introduce
a pair of "axes" so that we can associate coordinates to these points.
Finally, we observed that if the coordinates of the points are given by 
, then the coordinates of 
are given by
|
|
This is the basic law of vector addition.
We asked you to think
about the fact that if the axes (rulers) are changed, then the coordinates will
all change: 
and we will have
|
|
In other words, the
law of vector addition itself is independent of the choice of
axes (rulers). The addition law retains its form whatever coordinate
system we use. In more modern language, we would say that the addition law is
covariant.
We also define the so-called scalar product of a number by a
vector in the following way. If 
then if 
is a vector we say that
|
|
This definition,
like that for vector addition, is also covariant, that is, independent
of the choice of axes. Depending on the size and sign of 
it describes an expansion, contraction, or inversion of 
.
Since we considered
all choices of axes to be equivalent from the perspective of the algebra of
vectors, you can see that no covariant sense can be attached to
such common ideas as the angle between two vectors or the distance
between two points without further assumptions on the choice of axes. These
assumptions will specify the Geometry of the Euclidean plane.
Of course, we could
specify one particular set of axes and so one particular assignment of coordinates
to each vector in the plane, and then define distance and angle using those
coordinates. This is unsatisfactory. In fact, it is the opposite of covariance.
The geometric approach is to discover an underlying structure in the plane,
and then to consider all transformations of the plane that preserve that structure
to be "equivalent." The underlying structure should be preserved by
all parallel translations, for example, because these are the transformations
that define vectors in the first place. That structure is what we will call
the metric.
As
an aside, we point out that this geometric approach is at the heart of Einstein's
4-dimensional theory of gravity, for which the (local) transformations are the
ones that preserve "causality" in terms of light signals. These are
called Poincare transformations, or the more restricted Lorentz transformations.
As it turns out, the metric structure of the plane (and later, of space) can be specified with a simple extension of Pythagoras' theorem. We give the definition now:
Inner product (or dot product)
Definition 1: Suppose that in some system of coordinates (axes) 
are two points. Then define the inner product (also called the
"dot" product) of those points to be the number
|
|
This assigns to
each pair of vectors, a number, but the assignment obviously depends on the
system of coordinates, or axes. We want a structure that does not depend on
the particular choice of axes, and so we abstract away that choice by discovering
the basic properties of this inner product.
Question 1: Show that if, in some fixed system of coordinates, the inner
product is defined as above: for 
,
|
|
then the following statements are true:
Any rule for pairing
vectors with numbers that satisfies the four conditions above is called an inner product pairing. Notice that it does not necessarily depend
upon a particular choice of coordinates. We show that after we explain what
it means to "change coordinates."
Suppose given a pair
of axes (ruler x and ruler y) with respect to which coordinates
of a vector 
may be determined:
Given a second pair
of axes with the same common origin O (ruler 
and ruler 
) then the same point P will have different coordinates:
as below
The
question is, what is the relationship between the coordinates 
and 
? We can actually answer that question fairly easily. Suppose that the unit
vector of the 
is represented in x-y coordinates 
. That is:
|
|
and suppose that the other unit vector of the 
is represented in x-y coordinates 
.
That is:
|
|
Passive coordinate transformations
In this way, the
relationship between the coordinate system 
and the coordinate system 
is given by 4 numbers: 
.
In 
coordinates, we will represent the relationship by two column vectors
|
|
We use column vectors for reasons that will be clear in a moment.
Now if the vector
P has 
coordinates
|
|
then it is clear that in x-y coordinates,
|
|
We may write this:
|
|
Now since in x-y coordinates 
is also equal to
|
|
it follows that we have the pair of equations showing how to calculate the
coordinates of P from the 
coordinates:
|
|
and
|
|
Such mappings arise
frequently when we work with vectors. Let us abstract away from the meaning
of this transformation for a moment, and look at its structure. Suppose we represent
points in 
simply as columns (pairs of numbers)
|
|
without committing ourselves to any particular set of axes. These are simply algebraic objects that would acquire geometric interpretation if we chose axes. Still, we know how to add them, and to multiply them by scalars.
Such an algebraic
structure is called a vector space. In this light, the plane 
is a vector space. Now, consider a mapping T of the plane to itself
given by four numbers, 
and mapping
|
|
by the rule
|
|
This mapping has
the same form as the coordinate transformation rule just calculated. But as
an algebraic mapping, it has some simple properties that are easy to prove:
Question 2: Show that if X, Y are elements of 
and 
then
Also
show that if any system of axes whatsoever is used to correspond these columns
of numbers to vectors in the plane, then T carries straight lines
to straight lines, or to a single point. For a pair points U and V
with 
a straight line is defined as the set of points
|
|
for 
. Give an example of T for given 
that maps one straight line to a point, and a different straight line to a line.
Definition 2: These functions from 
determined by four numbers 
are called Linear Transformations. The linear transformation T
given by the rule
|
|
is represented by the 
array of numbers
|
|
It is called a 
matrix. The columns have the interpretation:
|
|
Since
|
|
it follows from the 3 properties of linear transformations above that
|
|
Now when a system
of axes is introduced in the plane with unit vectors, say, 
then if each vector is represented in the corresponding system of coordinates,
so, for example:
|
|
then a linear transformation T is a mapping from vectors to vectors, represented in this system of coordinates. Clearly,
|
|
and in general,
|
|
where
|
|
Now
suppose that a new set of 
are introduced. Suppose that the unit vector of the 
is represented in x-y coordinates 
, that is, 
,
|
|
And suppose that the unit vector of the 
is represented in x-y coordinates 
,
that is:
|
|
Then if the vector
P has 
coordinates
|
|
that same vector P has 
coordinates 
where, as we showed
|
|
We summarize this
by saying that when a fixed system of axes is introduced in the plane with unit
vectors: 
then vectors have coordinates in this system:
|
|
Matrix of a Linear Transformation
Now,
if a new system of axes is introduced, say 
then there is a unique linear transformation T such that
|
|
If,
when represented in 
coordinates, it has matrix
|
|
then this matrix also represents the change of coordinate formula from 
coordinates to 
coordinates. A vector 
with coordinates 
in the 
representation has coordinates 
in the 
representation.
In this way, we
see that a 
matrix represents at the same time, a linear transformation in the 
representation, and a change of coordinates from the 
representation to the 
representation.
Euclidean structure in the plane
We introduced this
notation in order to learn which linear transformations in the 
representation preserve a given inner product structure. That class of transformations
will define the Euclidean geometry for the plane. Thus, suppose that a set of
unit vectors 
is fixed. And suppose that an inner product 
is defined that satisfies the conditions:
The conditions 5,6, and 7 simply guarantee that if 
in the 
representation then
|
|
They guarantee that the system of unit vectors 
is what is called orthonormal.
We
will characterize the new systems of axes 
with the property that when this inner product is calculated in the new system,
it gives the same result as when it is calculated in the original. In fact,
in light of what we have just said, suppose that a new system of axes is introduced,
say 
. Then there is a unique linear transformation T such that
Active Coordinate Transformation
|
|
Suppose
that, when represented in 
coordinates, it has matrix
|
|
so that
|
|
and
|
|
Theorem 1: Under the above circumstances, the representation of the
inner product 
in the 
coordinates is given by the following formula: If 
in the 
representation, that is
|
|
and
|
|
then
|
|
Proof:
In the 
representation.
|
|
|
|
And so
|
|
Now expanding this expression using properties (1) -- (7) of the inner product
gives the result.
End of Proof
We see therefore
that if 
in the 
representation so that
|
|
then for the transformation T with matrix 
in the 
representation, 
and in particular,
|
|
So the change of axes T preserves the inner product if and only
if for all 
,
|
|
This means that the
matrix for T must satisfy the conditions:
1)
= 1
2)
= 1
3)
= 0
From the first two
equations, we see that 
are points on the unit circle. If 
and so from equation 3, 
and it follows from equation 2 that 
. So the four possible matrices for T in the case that 
are
|
|
In the case that 
we can conclude from the above that 
and so we can write
|
|
Thus, the line through
and 
is perpendicular to the line through 
and 
. It is not difficult to see that in any case there is an angle 
with
|
|
From that, we find that for this angle, there are four possible matrices for T
|
|
Now if we choose
the conventional axes with unit vectors
|
|
we can interpret the various choices of axes in the list of four above:
: rotate 
through the angle 
: rotate 
through the angle 
: map 
then rotate through the angle 
:
map 
then rotate through the angle 
In
the general case, we cannot talk of perpendicularity (yet), and so the transformation
of the first type would simply give
|
|
and
|
|
and so on. The transformations of type (3) and (4) "reverse orientation" and so in general are not physically meaningful.
Our conclusion is
that once an inner product structure is established for a some system of coordinates
in the plane satisfying conditions (1) -- (7) above, then it is covariant
for any changes of coordinates that rotate 
as in the equation just above. It has precisely the same form in any such rotated
system. In a moment, we will use this abstract inner product itself to define
the Euclidean geometry of the plane. After that, we will be justified to speak
of the "angle" between two vectors, and to assert that the axes are
perpendicular, and that the unit lengths along each axis are the same.
The point is that
we first need an inner product structure before we can define these ideas. And
it will follow that these definitions will be meaningful for all covariant (in
this case, rotated) systems of axes.
Suppose now that
we have an inner product 
that satisfies the conditions:
We
will now describe the geometry of the plane in terms of it. This plane-with-geometry
is sometimes called the Euclidean plane. This is to distinguish
it from the plane in which there is no additional inner product structure. In
the latter, there is no invariant notion of distance, angle, and so on. There
are only lines and points.
The
first thing that we will do is define the length of a vector, and the distance
between two vectors.
Definition 3: If A is a vector, then define the length of A
|
|
We see from condition 4 that if A is different from the 0 vector,
then it has positive length. If A and B are vectors,
define the distance from A to B to be the length
of the difference between them: 
Next, we define
the notion of perpendicular vectors.
Definition 4: Suppose that A and B are vectors.
Say that A is perpendicular to B ( 
) if 
.
If A and B are different from 0, then if they are
perpendicular, it cannot be the case that 
because this would imply that
|
|
and 
cannot be 0 because if it were 
would equal 0.
We next establish
an important inequality that will allow us to define the angle between two vectors,
the triangle inequality, and the notion of perpendicular projection.
Suppose that 
is a vector, and suppose that 
is any other vector. Construct the vector
|
|
Notice that
|
|
Expanding the right
hand side, we see that
|
|
or
|
|
thus
|
|
This is the Cauchy-Schwartz
inequality and taking square roots of both sides, we see that
|
|
It is easy also to conclude that
|
|
so that in fact,
|
|
We can have equality
only if 
which implies that 
,
that is V is a multiple of U.
Now from this inequality it is easy to establish the triangle inequality:
|
|
Just calculate
|
|
and calculate
|
|
This gives the result.
From this, we can conclude that if U, V, and W are vectors then
|
|
which is perhaps the more familiar form of the triangle inequality.
Orthogonal projection of one vector on another
Now, notice that the vector 
is actually perpendicular to U. A simple calculation shows that
|
|
Thus, writing
|
|
gives the unique resolution of V as a sum of vectors: W perpendicular
to U and 
parallel to U.
Question 3: Prove the resolution is unique. That is, if
|
|
then show that 
Gram-Schmidt Orthonormalization
Question 4: Now introduce a pair of axes with unit vectors 
show using the perpendicular projection result above that you may replace this
system with a system of unit vectors 
that satisfies conditions:
This is called Gram-Schmidt Orthonormalization. From now on we will work with such orthonormal axes with respect to the inner product.
Suppose then that
we have an orthonormal set of unit vectors 
.
And suppose that 
is any pair of non-zero vectors. We know that 
Therefore,
|
|
We will call the
angle between 0 and 
that satisfies
|
|
the angle between X and Y. Motivation for this is as follows. Suppose X and Y each have length 1. Write in our orthonormal system:
|
|
Then
|
|
Given a 
we know that it represents the linear transformation T with the
property that
|
|
We will refer to
the "standard" axes (with respect to our orthonormal system) in the
following way:
|
|
So the columns of the 
are the images under the linear transformation T (that the matrix
represents) of the standard vectors 
. In this way, we may identify a linear transformation ( 
) as an ordered pair of vectors.
Now suppose we represent
an arbitrary 
in polar form:
|
|
This represents a pair of vectors of lengths 
respectively. We know that the area of the parallelogram that these vectors
span is
|
|
This number is a
geometric property of the pair of vectors. If we rotate the axes, we will not
change the number. Now perform the following operation on the 
.
Calculate
|
|
Using the standard trigonometric identity, we see that this is equal to
|
|
Definition 5: Given 
,
the number 
is called the determinant of the matrix. We see that up to sign,
it is equal to the area of the parallelogram spanned by column vectors 
. Another interpretation of this number is that it is the ratio of the signed
area of the parallelogram spanned by 
to the unit area (1) of the square generated by standard basis vectors 
. In this view, the "sign" is positive if the transformation preserves
the orientation, and is negative if it reverses the orientation. Since the latter
property is also geometric, we see (or could prove directly) that the determinant
of a matrix is covariant: it remains the same under any rotation of axes.
The determinant
of a linear transformation is the final geometric invariant that we will study
for plane vectors. All of these things will be generalized to three dimensions
in the next part (Vectors in 3-Dimensions and Space Curves) but we will
conclude with an important fact about linear transformations and their determinants.
Determinant of a linear transformation of the plane
Suppose given two
linear transformations of the plane: T and S. Recall
that this means that if X, Y are elements of 
and 
then for transformation T
We are not assuming
that the transformations preserve the inner product structure. But they do map
lines to lines or points.
Question 5: Show that the sum of S with T:
is also a linear transformation. Next show that the composition of S
with T: 
is also a linear transformation. In particular, if in some Euclidean
system of axes, the matrices are:
|
|
End of Question
What is the matrix for 
? We have to calculate that. Certainly,
|
|
And
|
|
Thus,
|
|
|
|
By a similar calculation,
|
|
Therefore, the matrix
for the composition is
|
|
This defines the (associative) law of matrix multiplication
Definition 6: Given two linear transformations of the plane: S
and T: represented by 
then the product of the matrices is the matrix that represents the composition
. The formula is:
|
|
This multiplication is associative (because compositions of functions are
associative), but not commutative. The "identity" matrix 
is the multiplicative identity for this operation, just as the "zero"
matrix 
is the zero under addition. With the operations of addition, multiplication,
and "scalar multiplication" the set of 
form what is called a real algebra.
Recall the association
(determinant) that takes 
to real numbers.
|
|
We will call it det. This association has the remarkable property that
|
|
While it is reasonable,
it is by no means obvious. We could of course, prove it directly from the relation
already established:
|
|
but this would bypass many of the important properties of the determinant function that we will, in any case, need in 3 dimensions. Therefore, we will ask you to prove the following basic properties of det, and then we will derive the result from those, following the excellent book Advanced Calculus, by Nickerson, Spencer, and Steenrod.
Question 6: Considering det to be a real-valued function on pairs of column vectors:
|
|
show the following :
These properties
characterize the det function as an alternating, 2-linear function from
to real numbers.
Now consider the
matrix
|
|
Thinking in terms of column vectors, we may write this as the pair of columns:
|
|
Then using the multilinearity properties of det that you proved above, you see that
|
|
The first two terms are zero by property (5). Using property (4), we are left with:
|
|
which proves the result. This concludes the general discussion of the geometry of plane vectors.
We mentioned that
the set of 
form what is called a real algebra. To get some experience with
this algebra, we will consider in the exploration, the following subalgebra
of these matrices.
Question 7: Consider the set of 
of the form 
for 
. Show that if you add two of these, or if you multiply two of these, you get
a matrix of the same form. Show that multiplication is commutative for
matrices of this form. Show that if we consider [1] to
be "identity" matrix 
then the matrix [i] = 
belongs to this set, and that 
!
Thus, we can write each matrix in this set in the form:
|
|
As we will see below, this gives a representation of the complex numbers
(the Euclidean plane and more) as an algebra of matrices. From this point of
view, there is nothing mysterious about the equation 
. We have exhibited a solution.
Finally, show that if 
then 
. In particular, if either 
is different from 0, then 
and if we call the conjugate of 
|
|
then 
and so
|
|
or 
is the inverse of 
.
End of Question
Exploration: Geometric Invariants of Plane Vectors
Our
Cartesian plane has no pre-assigned axes. So if we want to associate coordinates
to points, the first thing to do is to create a pair of axes. You can create
two unit vectors, a blue one and a red one, by pressing the two buttons:
You
should probably begin with Integers checked: 
since this will select
the closest lattice points to the points you pick. After pressing each button,
click somewhere in the window away from the center. You
will see something like:
This
is all that is needed to associate coordinates with points in the plane. We
call the blue vector 
and the red vector 
. Now if you choose a point 
in the plane by pressing
and then clicking somewhere in the plane, the system will draw a black dot at the point you click on, and will draw a coordinate grid. Finally, it will report the (color-coded) coordinates of the point in the UV system, showing how the coordinates are obtained by the parallel postulate, as we explained. The parallel projections to the coordinate axes are also marked by a blue dot and a red dot..
In this case, the 
coordinate of 
is 0.2757142, and the 
coordinate of 
is 1.8571428.
Next,
if you click Show Unit Span, you will see the parallelogram spanned by
and 
.
You will also see a mysterious thing. You will see the determinant of a certain matrix. What matrix is that?
Well,
that is very much the point of this exploration. We assume that there is an
underlying inner product (Euclidean) structure in this plane. So vectors have
well-defined lengths, and have definite angles, and so on. In particular, if
we choose an orthonormal basis for this plane, then the pair of vectors 
and 
would be the columns of a 
matrix that represents them in terms of the orthonormal frame. The determinant
reported is a geometric invariant. It does not matter which orthonormal basis
we choose, as long as it is oriented consistently with our original one. Otherwise
the sign of the determinant would change.
Now,
we have two ways to see this. First, if you click one of the buttons:
you will see a new picture, and information printed in the MathEdit:
The
area is equal to the determinant in this case. It will always be equal to the
absolute value of the determinant. If you click the other button, you will see
the other projection, and the area reported will be the same.
Obviously, the area spanned by the vectors 
and 
must be a geometric invariant.
But
there is another way to see that. The entries in the 
matrix
represent the complex number 
as we mentioned above. When points in the plane are thought of as complex numbers,
then multiplication by this matrix amounts to rotation by 45 degrees in the
counter-clockwise sense.
You
may place any values you like in the matrix, but only rotations (possibly) combined
with inversions will preserve the geometric structure, as we saw above, those
will be matrices of the form:
Now,
you may apply the matrix as linear transformation in the UV frame to
the vectors 
and 
to get new versions of 
and 
.
Press Clear first, then press Apply the Matrix:
Now, when you Show Unit Span you will see:
This is a 45 degree rotation of the original pair. The determinant is still 7 units. Also, if you Project U onto V, you will see
These
are the same numbers as we saw in that operation above. The rotation preserves
the geometric properties of the plane. To convince yourself of this, try some
matrices that do not preserve the metric.
Now,
on the right hand side of the screen, we explore the relation between two different
choices of axes. Unless the new choice differs from the original one by a metric
preserving transformation, this has nothing in particular to do with geometry.
Still, you may create a new set of axes (called 
, now green and magenta) by clicking the
buttons then clicking the screen as before. If you click
you will see the new grid and the new coordinates for the
same point 
that you chose before. In this way, you may see how the UV coordinates
of a point are related to the U'V' coordinates of the same point. But
all of this is summarized if you press
For
then you will see the matrix transformation that carries UV coordinates
to U'V' coordinates.
.
