Inverse Quaternions

Date: 12/01/1999 at 06:09:16
From: Jarno Kuoppamaki
Subject: Inverse Quaternions

How do you calculate inverse quaternions? For example, the inverse of 
3 - 4i + 5j + 6k.

Date: 12/01/1999 at 12:27:40
From: Doctor Rob
Subject: Re: Inverse Quaternions

Thanks for writing to Ask Dr. Math, Jarno.

Call a = 3 - 4*i + 5*j + 6*k. Write the inverse in the form 1/a. 
Multiply the numerator and denominator by a with -i substituted for i, 
-j substituted for j, and -k substituted for k, or 

     3 + 4*i - 5*j - 6*k

Expand and simplify the denominator, which should result in a real 
number, 86 in this case: 

     86 = 3^2 + (-4)^2 + 5^2 + 6^2

Divide that real denominator into each coefficient in the numerator. 
Result:

     1/a = 3/86 + (4/86)*i - (5/86)*j - (6/86)*k

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/