THE SCALEARPRODUCT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

THE
SCALEARPRODUCT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

POSTULATES!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!:

1. THE SCALEARPRODUCT ARE FUNCTION: VECTOR X VECTOR - SCALEAR:
U X V -
U,V!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!

2. THE SCALEARPRODUCT ARE EULIDEAN
GEOMETRIC!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

""EXPLAINATION":
IF I DRAW THE TWO VECTORS, THEN THE SACALEAR PRODUCT ARE DETERMINED
FROM THE FIGURE DRAWN.
THAT IS: THE SCALEAR PRODUCT ARE FUNCTION OF |U|,|V| AND THE ANGLE
BETWEEN THE TWO VECTORS.
THE ANGLE BETWWEN TWO VECTORS ARE THE MINIUM ANGLE BETWEEN ANY
TWO LINES CONTAINING EACH ONE OF THE VECTORS.
THE 0VECTOR IS CONTAINED IN ANY LINE, SO THE ANGLE
BETWWEN ANY VECTOR AND THE 0VECTOR IS 0.
THE SYMBOLE OF ANGLE ARE v."
3. THE SCALEARPRODUCT ARE LINEAR IN BOTH
VECTORVARIABLES!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!

4. THE SCALEARPRODUCT OF
U,U=|U|^2!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!

LET THE TWO VECTORS BEE U AND V
IF ONE VECTOR OR BOTH ARE THE 0VECTOR, THE SCALEAR PRODUCT
ARE 0 FROM LINEARITY:
0=0U,V=0U,V=0,V
THE SAME FOR U.


LET NO ONE OF THE TWO VECTORS BEE THE 0VECTOR.

GIVEN THE UNIT VECTOR IN THE U DERECTION:
e
U=|U|e

IF U AND V ARE PARALLELL:
V=u|V|e

u ARE +1 or -1.

THEN:

U,V = |U|e,u|V|e = u|U||V|e,e = u|U||V||e|^2 = u|U||V|

IF U AND V ARE NOT PARALLELL
THEN THEY LIE IN ONE DETERMINED PLANE.


LET f BEE THE UNIT VECTOR PERPENDICULAR TO e IN THAT PLANE
LYING ON THE SAME SIDE OF THE ULINE AS V.

U=|U|.e.
V=|V|.[e.COS(v)+f.SIN(v)]

U,V = |U|.e, |V|.[e.COS(v)+f.SIN(v)]
=|U||V|.e,e.COS(v)+ |U||V|.e,f.SIN(v)
=|U||V|.|e|^2.COS(v)+ |U||V|.e,f.SIN(v)
=|U||V|.COS(v)+ |U||V|.e,f.SIN(v)


FORMULA THE SCALEARPRODUCT:
U,V = |U||V|.COS(v)+ |U||V|.e,f.SIN(v)

AS YOU CAN SEE, THIS FORMULA
IS 0 IF ONE OR BOTH VECTORS ARE THE 0VECTOR
AND IF THEY ARE PARRALLELL THE SECOND TERM VANISCH
GIVING only |U||V|.COS(v)
AND THAT IS THE SAME AS u|U||V|,
BOTH IF u=+1 : COS(v)=1 (THEY LIE ON THE SAME SIDE)
AND IF u=-1 : COS(v)=-1 (THEY LIE OOPPOSITE)
SO THE FORMULA IS GENERAL:

FORMULA THE SCALEARPRODUCT:
U,V = |U||V|.COS(v)+ |U||V|.e,f.SIN(v)

I MUST PROOF THAT THE e,f = +0

e,f ARE THE SCALEARPRODUCT OF TWO ORTOGONAL UNIT VECTORS,
AND POSTULATE 2 SAY THAT IT IS DETERMINED.
f,e = e,f FROM POSTULATE 2

PROOF THAT e,f=0:
LET TWO VECTORS BEE:
W=|W|.[e.COS(vW)+f.SIN(vW)]
X=|X|.[e.COS(vX)+f.SIN(vX)]
LET |W| AND |X| NOT BEE 0
LET 180 DEGREE vX vW 0


e AND f are here TWO ORTOGONAL UNIT VECTORS.

W,X = |W||X|
[[|e|^2 COS(vW) COS(vX) + |f|^2 SIN(vW) SIN(vX)]
+e,f COS(vW)SIN(vX) + f,e SIN(vW) COS(vX)]

=|W||X| COS(vX-vW)
+|W||X|.e,f.SIN(vX+vW)

BUT FORMULA THE SCALEARPRODUCT:
W,X = |W||X|.COS(v)+ |W||X|.e,f.SIN(v)

v is here the ANGLE BETWEEN W AND X
v = vX - vW

W,X = |W||X|.COS(vX - vW)+ |W||X|.e,f.SIN(vX - vW)

I NOW HAVE TWO EXPRESSION OF W,X, EAUALITY YELDS:

|W||X| COS(vX-vW)+|W||X|.e,f.SIN(vX+vW)
=|W||X|.COS(vX - vW)+ |W||X|.e,f.SIN(vX - vW)

|W||X|.e,f.SIN(vX+vW)
=|W||X|.e,f.SIN(vX - vW)

DIVIDIE WITH |W|,|X|


e,f.SIN(vX+vW)=e,f.SIN(vX - vW)

e,f.[SIN(vX+vW)-SIN(vX - vW)]=0
e,f.[ COS(vW)SIN(vX) + SIN(vW) COS(vX)
-( COS(vW)SIN(vX) - SIN(vW) COS(vX))]=0

2.e,f.SIN(vW) COS(vX)=0
LET vW BEE 90 DEGREE:
2.e,f COS(vX)=0
BUT 180 DEGREE vX vW 0
BUT 180 DEGREE vX 90 DEGREE 0
AND COS(vX) ARE NOT 0 IN THAT INTERVALL:

e,f=0
Q.E.D.


FORMULA THE SCALEARPRODUCT:
U,V = |U||V|.COS(v)+ |U||V|.e,f.SIN(v)

FORMULA THE SCALEARPRODUCT:
U,V = |U||V|.COS(v)+ |U||V|.0.SIN(v)

"""""""""""""""""""""""""FORMULA
THE
SCALEARPRODUCT:"""""""""""""""""""

"""""""""""""""""""""""U,V
=
|U||V|.COS(v)""""""""""""""""""""""""""

I HAVE PROOFED THIS FORMULA FROM THE POSTULATES.
THE POSTULATE 1,2 AND 4 IS TRIVIAL FROM THIS FORMULA.

HOW TO PROOF POSTULATE 3: LINEARITY: FROM THE
FORMULA??????????????????????????????????????????? ?????????????????????????????????????????????????? ?????????????????????????????????????????????????? ?????????????????????????????????????????????????? ?????????????????????????????????????????????????? ?????????????????????????????????????????????????? ?????????????????????????????????????????????????? ?!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!!!