Is quaternion multiplication associative?

Hi,

I know that quaternions hold associative property,

which is A.(B.C) = (A.B).C

But, have a look at the code below

%%

t1 = sym('t1/2','real');

t2 = sym('t2/2','real');

t3 = sym('t3/2','real');

Q01 = [cos(t1) 0 0 sin(t1)];

Q12 = quatmultiply([cosd(45) sind(45) 0 0],[cos(t2) 0 0 sin(t2)]);

Q23 = quatmultiply([cosd(-45) sind(-45) 0 0],[cos(t3) 0 0 sin(t3)]);

X = quatmultiply(Q01,quatmultiply(Q12,Q23));

Y = quatmultiply(quatmultiply(Q01,Q12),Q23);

X-Y

%%

but there's a difference between 'X-Y'. How come?

can someone help me with this.

Thanks.

Answer 1

[James Tursa]

You just need to simplify it to see that the result is in fact 0's. E.g., running your code gives this for X-Y:

ans =

[ cos(t1/2)*cos(t2/2)*cos(t3/2) - cos(t2/2)*sin(t1/2)*sin(t3/2) - (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 - (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 - (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 + (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 - (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 - (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 + (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 + (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2, cos(t3/2)*sin(t1/2)*sin(t2/2) - cos(t1/2)*sin(t2/2)*sin(t3/2) + (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 - (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 - (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 + (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 + (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 - (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 + (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 + (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2, (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 - (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 - sin(t1/2)*sin(t2/2)*sin(t3/2) - cos(t1/2)*cos(t3/2)*sin(t2/2) + (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 + (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 - (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 - (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 + (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 + (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2, cos(t1/2)*cos(t2/2)*sin(t3/2) + cos(t2/2)*cos(t3/2)*sin(t1/2) + (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 - (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 - (2^(1/2)*cos(t3/2)*((2^(1/2)*cos(t1/2)*sin(t2/2))/2 + (2^(1/2)*cos(t2/2)*sin(t1/2))/2))/2 - (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 - (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2 - (2^(1/2)*sin(t3/2)*((2^(1/2)*cos(t1/2)*cos(t2/2))/2 + (2^(1/2)*sin(t1/2)*sin(t2/2))/2))/2]

>> simplify(ans)

ans =

[ 0, 0, 0, 0]