Question

6. Establish the following quaternion identities: (a) (21+ yii +213 + wik)(x2 - y2i - 22j - w2k) = 2102 + y1y2 + 2122 + WW2) + (-2192 + 9122 - 21W2 + wz2) +(-2122 + 2132 - Wiy2 + yi w2)j + (-21 W2 + w122 - Yi Z2 + ziy2)k (b) (x2 + y2i + z2j + wak)(x1 - yii - zij - Wik) = (20102 + 91 92 + 2122 + WW2) + (a192 - 9112 + 2qW2 - W122) i +(0122 - 2102 + wiY2 - y1w2)3 + (T1W2 - W122 + y122 - 2142)k

Answer 1

Solutions Given that we knows{t1; I 1, ti, zk} is group of quaternion satisfyins the following realation OP=j?k=-1 دا را = ال ر = ر jolak Keja-i iikaaj @ Cost Hoit 200+ wik) (dowy,i+zei work) = [X 2q - 7, 9, 1-4; 2j-t, uz k) + (9, 4, 1-9, 9, 12 9, Zij- y, cozink) + (2, 42) - Zgjila - 2q wa sik) + (W, Xak - ww y, k,i-wz kig - w, W2 K²)

= (xo de - Yo Y21 – 31221 – ,W2kJ+(9,821+9,95 9, 22k ty, wzj) +(2, 4) tz 92k +2122-2, 42.1) +(Wildak - W.J₂Jt Wizzito, wal calculating to like terms in the above we get = (op de ty, Yzt Zi2+ w ws)+(- & Y₂ typ 2₂ -2, W2 +Wazli+(-1,2₂ t9, W₂ + 2, W2 - wo yli +(-7, wz - Y, Y2 + 2,4, tur talk to first Thus we established indentity.

© (x2+ y2i+22it Waky (86-94-21j – Wik) =( No X2 -1 2-4, 1-22Z , j - 22 4 k) + (Yes,j-Yeyi"? y? ij - Yq W. (r) & (2, zgj ty, zk +2,22-W 223)+(8 Weke - Y, wuj tzwei'tlo ws] Now collecting the cléke Hernesweget = (a da ty, Y2+3,22+w, wz)+(-7, X2 + 41 Yy-61 23t2,wf? . + (-2,02 + y₂ + x1 Zz-y, Waj+ (-42W, Z,Y, +4,2 + 862) K Thus we establish the second identity