# why is representing rotations through quaternions more compact and quicker than using matrices??

The representations of rotations by quaternions are more compact and quicker to compute than the representations by matrices.

However, I have to admit, I don't fully understand what quaternions are and why they are useful. I have tried to read the article, but I don't understand why defining such a system is useful.

It appears to a define a four dimensional space in which 3 components are imaginary and one is real. Is this attempting to describe spacetime?

Regardless, I was hoping someone here could show how to represent a rotation using both quaternions and matrices and compare the two for me.

• Have you looked at the questions already answered on our site? The list of Related questions in the margin is a good starting point. The use of quaternions in representing rotations is covered in many of those. – Jyrki Lahtonen Jul 9 '15 at 13:31
• Quaternion algebra is used in many places in physics: SU(2) group is used in quantum mechanics and such. AFAICT it is not really to describe spacetime in spite of the apparent 3+1 dimensions. The metric of quaternions is Euclidean (as opposed to Lorentzian). Physicists can say something more precise. – Jyrki Lahtonen Jul 9 '15 at 13:35
• You can see a symilar question here. math.stackexchange.com/questions/435680/… – Emilio Novati Jul 10 '15 at 17:31
• Quaternions are always more compact: it uses fewer numbers, and then presumably less computation. – rschwieb Jul 16 '15 at 15:38

• I'm assuming you are thinking about a rotation matrix in 3D. You can get away with storing a vector $\bf u$ and an angle $\theta$ which makes for 4 values, but then you have to do more cumbersome calculations building several matrices on the fly and using linearity of matrix multiplication. – mathreadler Jul 9 '15 at 15:19