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Quaternion * Quaternion = Quaternion
(a+bi+cj+dk)*(w+ xi+ yj+ zk) = (aw-bx-cy-dz) +(ax+bw+cz-dy)i +(ay-bz+cw+dx) j +(az+by-cx+dw)k
a, b, c, d, w, x, y, z
?Real
b. Assosiatif
(a+bi+cj+dk)*((w+ xi+ yj+ zk)*(e+ fi+ gj+hk)) = ((a+bi+cj+dk)*(w+ xi+ yj+ zk))*(e+ fi+ gj+hk)
Bukti :
(w +
xi +
yj +
zk ) *
(e + fi +
gj + hk )
=
(we –xf-yg –zh) + (wf +xe + yh – zg)i +(wg –xh
+ye +zf)j + (wh + xg – yf + ze)k
(a + bi + cj + dk ) *
((w +
xi +
yj +
zk ) *
(e + fi +
gj + hk
)) =
=( a (we-xf-yg-zh) – b(wf+xe+yh-zg)-c(wg-xh+ye+zf) –d(wh+xg-yf+ze)) +
(a (wf+xe+yh-zg) + b(we-xf-yg-zh) + c(wh+xg-yf+ze) –d(wg-xh+ye+zf))i +
(a(wg –xh+ye+zf) –b (wh+xg-yf+ze) +c (we-xf-yg-zh) +d(wf+xe+yh-zg))j +
(a(wh+xg-yf+ze) + b ( wg-xh+ye+zf)- c(wf+xe+yh-zg) + d(we-xf-yg-zh))k
=(awe –axf-ayg-azh –bwf-bxe-byh+bzg –cwg+cxh-cye-czf-dwh-dxg+dyf-dze) +
(awf+axe+ayh-azg +bwe –bxf-byg-bzh+ cwh +cxg –cyf +cze –dwg +dxh –dye –dzf)i
(awg-axh+aye+azf-bwh-bxg+byf-bze +cwe-cxf-cyg-czh + dwf+dxe +dyh-dzg)j
(awh+axg-ayf+aze + bwg-bxh+bye+bzf –cwf-cxe-cyh+czg +dwe-dxf-dyg-dzh)k
(a + bi + cj + dk ) *
(w + xi +
yj +
zk
)
=
(aw –bx-cy –dz) + (ax +bw + cz – dy)i +(ay –bz
+cw +dx)j + (az + by – cx + dw)k
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