Mathematics 138
Vectors, Doublets, and Duality
A Galilean vector v may be specified by an
ordinary line and an ordinary distance. The 
vector can be represented as the directed
segment between two points A and B on the
line that are the specified distance apart.
A is called the "tail" of the vector and
B is called the "head."
A different pair of points A' and B' on the
same line can specify the same vector.
A Galilean doublet V may be specified by a point and an angle measure. The 
doublet can be represented as the directed
angle between two lines a and b through the point that have the specified angle between them.
A is called the "beginning" of the doublet and B is called the "end." 
A different pair of lines a' and b'  through the same point can specify the same doublet.

Using a different line parallel to the first line
can represent the same vector.  Just use points
A" and B" on the new line that are parallel to A
and B.
Using a different point parallel to the first point can represent the same doublet.
Just use lines a" and b" through the new point that are parallel to a and b.

Thus any line in the plane can be the
"beginning" of a doublet

Thus any point in the plane can be the "tail" of
a vector.

A vector that is specified by two parallel
points is a special vector.  The special 
distance between the points is the special
length of the vector.
A doublet that is specified by two parallel
lines is a special doublet.  The special 
distance between the lines is the special
length of the doublet.

Any other pair of parallel lines with the
same special distance specifies the same
doublet.  This other pair of lines can be
parallel to the original pair
on in a different direction making some 
angle with the original

Any other pair of parallel points with the
same special distance specifies the same
vector.  This other pair of points can be
on the same special line as the original pair
on or a different special line some ordinary
distance from the original.

It is easy to add two vectors v and w if the
 "head" B of the first one coincides with the
 "tail"  B of the second one.  
If v is represented by AB and w by BC, the
sum of the two vectors is represented by AC. 
It is easy to add two doublets V and W if the
 end B of the first one coincides with the
 beginning  B of the second one.  
If V is represented by ab and W by bc, the
sum of the two doublets is represented by ac. 

If two doublets are drawn "beginning to
beginning" so that
V is represented by ab and W by ad, then
they are added by the coparallelogram law.
Just replace ad by an equal doublet bc.
Now they are end to beginning..

If two vectors are drawn "tail to tail" so that
v is represented by AB and w by AD, then
they are added by the parallelogram law.
Just replace AD by an equal vector BC.
Now they are head to tail.

Or you could replace AB by DC and get
the same answer.  The four points ABCD
are the vertices of a parallelogram.
Or you could replace ab by dc and get
the same answer.  The four lines abcd
are the sides of a coparallelogram.

To multiply a doublet by a real number a, leave its beginning fixed and move the end so that the angle (or special distance) from beginning to end is multiplied by a.

To multiply a vector by a real number a, leave
its tail fixed and move the head so that the
distance (or special distance) from head to
 tail is multiplied by a.

As a basis for the vector space, choose
any ordinary vector i with length 1 and
the special vector o with special length 1.
Customarily these two vectors are drawn
so that their tails coincide, and the common
point that is the tail of each is the origin.
As a basis for the vector space, choose
any ordinary doublet I with angle 1 and
the special doublet O with special length 1.
Customarily these two doublets are drawn
so that their beginnings coincide, and the common point that is the beginning of each
is the horizontal axis.

Now any vector v in the Galilean plane can
be represented as a linear combination
v = xi + yo.  The usual way to show this
is take the tail of v at the origin A and
to translate the vector yo to a point B
that coincides with the head of xi.
Then, if the head of yo is point C, the
vector v runs from A to C.
Now any doublet V in the Galilean plane can
be represented as a linear combination
V = xI + hO.  The usual way to show this is to take the beginning of V as the horizontal axis a and
to translate the doublet hO to a line b that coincides with the end of xI. Then, if the end of hO is line c, the doublet V runs from a to c.

So any vector v can be represented by
an element of R2, namely . In this vector
space there is an inner product, defined by
<,> = x1 x2, with the property
that the inner product <,> = x2
is never negative. 
The vector is perpendicular to every vector.
The multiples of this vector form a 1-dimensional
Euclidean vector space with length defined
by special distance.
So any doublet V can be represented by
an element of R2, namely . In this vector space there is an inner product, defined by
<,> = x1 x2, with the property
that the inner product <,> = x2
is never negative. 
The vector is perpendicular to every vector. The multiples of this vector form a 1-dimensional Euclidean vector space with length defined by special distance.
Any point of Galilean geometry can be
represented in a non-unique way
by a line m and a doublet V. 
The doublet V is most easily chosen as mn
where n is some other line through the point.
Any line of Galilean geometry can be
represented in a non-unique way
by a point P and a vector v. 
The vector v is most easily chosen as PQ
where Q is some other point on the line.

The line is the set of points P + tv for
arbitrary real t.  All these points lie on the same line.
Algebraically the line is specified by
x = P1 + v1 t
y = P2 + v2 t
If we eliminate t, we have

which is a linear equation in x and y
satisfied by every point on the line.

Conclusions:
The system of points and vectors satisfies
all the axioms of Galilean geometry listed
in supplement B of Yaglom.
A line may by expressed in terms of
a point and a vector or analytically in
terms of the coordinates x and y of the
points that lie on it.
The point is the set of lines m + tV for arbitrary real t.  All these lines pass through the same point.
Algebraically the point is specified by
x = m1 + V1 t
h = m2 + V2 t
If we eliminate t, we have

which is a linear equation in x and h
satisfied by every line through the point
Conclusions:
The system of lines and doublets satisfies
all the axioms of Galilean geometry listed
in supplement B of Yaglom.
A point may by expressed in terms of
a line and a doublet or analytically in
terms of the coordinates x and h of the
lines that pass through it.

Check of the axioms for Galiliean geometry (see Yaglom Supplement B for details)
Groups I and II -- the usual vector-space axioms
These are satisfied either by vectors (made from points) or by doublets (made from lines)
Group III - dimension axioms
For either vectors or doublets, a basis for the vector space has two elements
Group IV - scalar product axioms
Either for vectors or for doublets, the scalar product <> = x1 x2 is commutative,
associative, and distributive. It is also semi-definite: <> cannot be negative. There
are nonzero elements (special vectors) or special doublets whose length is 0. So the
distinctive axiom IV5(G) is satisfied, either by vectors or by doublets:
"There exists a vector a with a2 > 0 and a nonzero vector o with o2 = 0."
Group V -- point vector relation
V1 - For every point (or line) A and vector (or doublet) a there exists there is a unique point B 
such that the vector (or doublet) from A to B equals a.

V2 - For any three points (or lines) A, B, C 
with u = vector (or doublet) from A to B
v = vector (or doublet) from B to C
w = vector (or doublet) from A to C
w = u + v
This was shown by introducing a basis, since the statement is clearly true when vectors
(or doublets) are represented by elements of R2
So, given the axioms, lines and doublets satisfy them just as well as do points and vectors.
This means that, from an axiomatic viewpoint, a formulation in terms of lines and doublets
is equivalent to one in terms of points and vectors.

Given any theorem that was proved from the axioms using points and vectors, we can construct the dual theorem about lines and doublets and prove it in exactly the same way from the axioms, treating the axioms as statements about lines and doublets.

A vector proof of the "parallel" case of Desargues theorem:

Let u = AA'      v = AB      w = AC
AA'B'B is a parallelogram so AB' = u + v
AA'CC' is a parallelogram so AC' = u + w
Thus both BC and B'C' = w - v and the lines BC and B'C' are parallel.
The diagram has five vectors whose tail is at A.
A statement and a doublet proof of the Galilean dual of this theorem.
Start with six lines a,b,c, a', b', c'.
Suppose that points aa', bb', and cc' are all parallel points (heavy vertical dotted line).
Suppose that points ab and a'b' are parallel( far right) and that ac and a'c' are parallel(far left)
Then points bc and b'c' (middle light dotted line) are also parallel.

Let doublet U = aa', doublet V = ab, doublet W = ac
aa'bb' is a coparallelogram so doublet ab' = U+V.
aa'cc' is a coparallelogram so doublet ac' = U+W.
Thus both doublet bc and doublet b'c' are equal to W-V and the points bc and b'c' are parallel.
The diagram has five vectors whos beginning is at a.

The dual of a dilatation is a compression
The simplest dilatation is one with the
origin A as its center and stretching factor k.
Every point P=is transformed 
into P' = . The point P' lies on the
line AP,
but the distance d(P',A) = k d(P,A).

The simplest compression is one with the
horizontal axis a as its axis and
stretching factor K
Every line m =is transformed 
into m' = . The line m' passes through
the intersection of a and m,
but the angle between m' and a (the slope of m')
is K times the angle between m and

More generally, choose the axis a to be an
arbitrary line.  The analytic representation of
the compression is now more complicated, but
is is still true that
the line m' passes through the point am
and the angle between m' and a
is K times the angle betwen m and a.

A compression multiplies all angles and special
distances by K, but ordinary distances
are invariant.
More generally, choose the center A to be an
arbitrary point.  The analytic representation of
the dilatation is now more complicated, but
is is still true that
the point P' lies on the line AP,
and the distance d(P',A) = k d(P,A).

A dilatation multiplies all distances and
special distances by k, but Galilean angles
are invariant.