# include <stdio.h>
# include <math.h>

int main ( )
{
  double a12;
  double a13;
  double a23;
  double c12;
  double c13;
  double c23;
  int i;
  double pi = 3.14159265;
  double v1[3] = { 1.0, 0.0, -1.0 };
  double v2[3] = { 4.0, 5.0,  4.0 };
  double v3[3] = { 4.0, 3.0,  0.0 };
  double v1_norm;
  double v2_norm;
  double v3_norm;
  double v1v2;
  double v1v3;
  double v2v3;
//
//  The length of a vector is the square root of the sum of the squares
//  of the entries.  Here are three ways to compute it.
//
  v1_norm = sqrt ( v1[0] * v1[0] + v1[1] * v1[1] + v1[2] * v1[2] );

  printf ( " ||v1|| = %g\n", v1_norm );

  v2_norm = sqrt ( pow ( v2[0], 2 ) 
                 + pow ( v2[1], 2 ) 
                 + pow ( v2[2], 2 ) );

  printf ( " ||v2|| = %g\n", v2_norm );

  v3_norm = 0.0;
  for ( i = 0; i < 3; i++ )
  {
    v3_norm = v3_norm + v3[i] * v3[i];
  }
  v3_norm = sqrt ( v3_norm );

  printf ( " ||v3|| = %g\n", v3_norm );
//
//  The dot product of two vectors is 0 if they are perpendicular to each other.
//  Here are two ways to do this.
//
  printf ( "\n" );
  v1v2 = v1[0] * v2[0] + v1[1] * v2[1] + v1[2] * v2[2];

  printf ( " v1.v2 = %g\n", v1v2 );

  v1v3 = v1[0] * v3[0] + v1[1] * v3[1] + v1[2] * v3[2];
  printf ( " v1.v3 = %g\n", v1v3 );

  v2v3 = 0.0;
  for ( i = 0; i < 3; i++ )
  {
    v2v3 = v2v3 + v2[i] * v3[i];
  }
  printf ( " v2.v3 = %g\n", v2v3 );
//
//  The cosine of the angle between two vectors is the dot product divided by their norms.
//
  printf ( "\n" );
  c12 = v1v2 / v1_norm / v2_norm;
  printf ( " cosine(v1,v2) = %g\n", c12 );
  c13 = v1v3 / v1_norm / v3_norm;
  printf ( " cosine(v1,v3) = %g\n", c13 );
  c23 = v2v3 / v2_norm / v3_norm;
  printf ( " cosine(v2,v3) = %g\n", c23 );
//
//  The ANGLE between two vectors is the inverse cosine or "arc-cosine" of the cosine.
//  This will come out in RADIANS unless we divide the result by PI and multiply by 180.
//
  printf ( "\n" );
  a12 = acos ( c12 ) / pi * 180;
  printf ( " angle(v1,v2) = %g degrees\n", a12 );
  a13 = acos ( c13 ) / pi * 180;
  printf ( " angle(v1,v3) = %g degrees\n", a13 );
  a23 = acos ( c23 ) / pi * 180;
  printf ( " angle(v2,v3) = %g degrees\n", a23 );

  return 0;
}