
 a) Verify with numeric applications some properties. 
    
 b) The coefficients, are randomly selected by the computer.

 c) The size of the vectors, are randomly selected by the 
    computer, but you can select the size if you want.

 d) F : Fraction version.
    I : Integer  version.

 
   Try the examples in this order.

    Properties of Euclidian inner product in R**n :
    =============================================

                         u.v  = v.u       : euinprdaF.c 
                   (u + v).w  = u.w + v.w : euinprdbF.c
                      (ku).v  = k(u.v)    : euinprdcF.c 
                         u.u >= 0         : euinprddF.c


    If u and v are vector in R**n with the Euclidean inner product : 
    ==============================================================
     
    u.v = 1/4 ||u+v||**2 - 1/4 ||u-v||**2 : thuvF.c


    Cauchy-Schwarz inequality in R**n :  
    =================================

                     |u.v| =< ||u|| ||v|| : causchF.c 


    If u and v are orthogonal in R**n with the Euclidean inner product  : 
    ==================================================================

         ||u+v||**2 = ||u||**2 + ||v||**2 : pythgF.c



    Properties of length in R**n :
    ============================

               ||u + v|| =< ||u|| + ||v|| : lngthdCF.c (triangle inequality)


    Properties of distance in R**n :
    ==============================

              d(u,v) >=   0               : distcaCF.c 
              d(u,v)  =   d(v,u)          : distccCF.c 
              d(u,v)  =<  d(u,w) + d(w,v) : distcdCF.c (triangle inequality)
