쿼터니언(Quaternion)

* magnitude 부분이 논란이 있으니 좀더 조사 要

 q=s+i*a+j*b+k*c is a quternion if
(1) i^2,j^2, and k^2=-1,
(2) ij=-ji=k
(3) jk=-kj=i
(4) ki=-ik=j
(5) s,a,b,c are real numbers (scalar).

q1+q1 = (s1+s2)+i*(a1+a2)+j(b1+b2)+k*(c1+c2)
for two quternion q1=s1+i*a1+ ... and q2=s2+i*a2+... .

d*q1=d*s+i*(d*a1)+...  for d is scalar.

q can be represented as q=(s,v) where v=(a,b,c) .. called ordered pair notation.

complex conjugate of q: (s,-v)
q*(complex conjugate of q): s^2+v·v
the magnitude of q : sqrt(q*complex conjugate of q)
NOTE: the above is the same as in complex number system.

inverse of q=(s,-v)/(2nd power of the magnitude of q)=(s,-v)/(s^2+v·v)
q*(inverse of q)=1 (becomes scalar) =(1, 0)

then q1+q2 = (s1+s2, v1+v2)
q1*q2 = (s1+i*a1+j*b1+k*c1)*(s2+i*a2+j*b2+k*c2)
          = s1*s2 +(i^2)*a1a2+(j^2)*b1b2+(k^2)*c1c2
             +s1(i*a2+j*b2+k*c2)+ s2(i*a1+j*b1+k*c1)
             +jk(b1c2-b2c1)
             +ki(c1a2-a1c2)
             +ij(a1b2-a2b1)
  = (s1s2-v1·v2, s1*v2+s2*v1+v1×v2)