Section 4: Dot and Cross Products


In R^3,

Input := 

Collect[(a d[x] + b d[y] + c d[z]) ^ 
        (e d[x] + f d[y] + g d[z]),
        {d[x]^d[y],d[y]^d[z],d[x]^d[z]}] 

Output =

(a d[x]) ^ (e d[x]) + (a d[x]) ^ (f d[y]) + 
 
  (a d[x]) ^ (g d[z]) - (e d[x]) ^ (b d[y]) - 
 
  (e d[x]) ^ (c d[z]) + (b d[y]) ^ (f d[y]) + 
 
  (b d[y]) ^ (g d[z]) - (f d[y]) ^ (c d[z]) + 
 
  (c d[z]) ^ (g d[z])

This gives the vector, or cross, product in vector algebra.

Input := 

Collect[(a d[x] + b d[y] + c d[z]) ^ 
        (e d[y]^d[z] + f d[z]^d[x] + g d[x]^d[y]),
        {d[x]^d[y]^d[z]}]

Output =

-(g d[x] ^ (a d[x]) ^ d[y]) + f d[x] ^ (a d[x]) ^ d[z] + 
 
  g d[x] ^ d[y] ^ (b d[y]) + g d[x] ^ d[y] ^ (c d[z]) + 
 
  f d[x] ^ (b d[y]) ^ d[z] - f d[x] ^ d[z] ^ (c d[z]) + 
 
  e (a d[x]) ^ d[y] ^ d[z] - e d[y] ^ (b d[y]) ^ d[z] + 
 
  e d[y] ^ d[z] ^ (c d[z])

This gives the inner, or dot, product in vector algebra.

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).