Quaternions

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Quaternions

Quaternions PDF

Sir William Rowan Hamilton(1805 – 1865) came up with the following revelation in dealing with extended complex number theory that you need to have 4 dimensions in order to rotate points in three dimensions:-

For complex numbers in two dimensional form:

q=w+ix; wherei=√(-1)

In four dimensions called Quaternions by Hamilton :-

q=w+ix+jy+kz

In a flash of inspiration Hamilton came up with the following which correctly observes all the properties of quaternions, he was so excited about it he carved this on a bridge in Ireland:-

 

i^2=j^2=k^2=ijk=-1

Properties of quaternions can be remembered with the aid of the diagram above:-

 

Starting with i and going in a clockwise direction:

ij=k

jk=i

ki=j

Starting with i and going in an anticlockwise direction:-

ik=-j

kj=-i

ji=-k

Note multiplication is not commutative!

 

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