A vector is defined as having three dimensions as being represented by an ordered collection of three numbers:   (X, Y, Z).

If you imagine a graph with the   x   and   y   axis being at right angles to each other and having a third,   z   axis coming out of the page, then a triplet of numbers,   (X, Y, Z)   would represent a point in the region,   and a vector from the origin to the point.

Given the vectors:

        A = (a1,  a2,  a3) 
        B = (b1,  b2,  b3) 
        C = (c1,  c2,  c3) 

then the following common vector products are defined:

 The dot product       (a scalar quantity)
  A • B = a1b1   +   a2b2   +   a3b3  
 The cross product       (a vector quantity)
  A x B = (a2b3  -   a3b2,     a3b1   -   a1b3,     a1b2   -   a2b1)  
 The scalar triple product       (a scalar quantity)
  A • (B x C) 
 The vector triple product       (a vector quantity)
  A x (B x C) 


Task

Given the three vectors:

        a = ( 3,    4,    5)
        b = ( 4,    3,    5)
        c = (-5,  -12,  -13)

 Create a named function/subroutine/method to compute the dot product of two vectors.
 Create a function to compute the cross product of two vectors.
 Optionally create a function to compute the scalar triple product of three vectors.
 Optionally create a function to compute the vector triple product of three vectors.
 Compute and display: a • b
 Compute and display: a x b
 Compute and display: a • (b x c), the scalar triple product.
 Compute and display: a x (b x c), the vector triple product.


References

   A starting page on Wolfram MathWorld is   Vector Multiplication .
   Wikipedia   dot product. 
   Wikipedia   cross product. 
   Wikipedia   triple product.


Related tasks

   Dot product
   Quaternion type


let a = (3.0, 4.0, 5.0)
let b = (4.0, 3.0, 5.0)
let c = (-5.0, -12.0, -13.0)
 
let string_of_vector (x,y,z) =
  Printf.sprintf "(%g, %g, %g)" x y z
 
let dot (a1, a2, a3) (b1, b2, b3) =
  (a1 *. b1) +. (a2 *. b2) +. (a3 *. b3)
 
let cross (a1, a2, a3) (b1, b2, b3) =
  (a2 *. b3 -. a3 *. b2,
   a3 *. b1 -. a1 *. b3,
   a1 *. b2 -. a2 *. b1)
 
let scalar_triple a b c =
  dot a (cross b c)
 
let vector_triple a b c =
  cross a (cross b c)
 
let () =
  Printf.printf "a: %s\n" (string_of_vector a);
  Printf.printf "b: %s\n" (string_of_vector b);
  Printf.printf "c: %s\n" (string_of_vector c);
  Printf.printf "a . b = %g\n" (dot a b);
  Printf.printf "a x b = %s\n" (string_of_vector (cross a b));
  Printf.printf "a . (b x c) = %g\n" (scalar_triple a b c);
  Printf.printf "a x (b x c) = %s\n" (string_of_vector (vector_triple a b c));
;;