Why is quaternion multiplication commutative?

The complex numbers are created from the real numbers by applying the Cayley–Dickson construction , and the quaternions are created from the complex numbers by a second application of the same technique. The multiplication defined involves taking conjugates, and that’s why quaternion multiplication isn’t commutative.

How do you multiply two quaternions together?

The multiplication rules for the imaginary operators are the same as for other numbers. We just put each quaternion in brackets and multiply out all the terms: (a + i b + j c + k d)*(e + i f + j g + k h). When we are multiplying the imaginary operators we use the following rules: i*i = j*j = k*k = -1.

Why do quaternions represent rotations?

Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo- metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.

What is the quaternion order?

The quaternions are a number system in the form a+bi+cj+dk. The quaternions ±1, ±i, ±j, ±k form a non-abelian group of order eight called quaternion group. Quaternion group can be represented as a subgroup of the general linear group GL2(C).

How do you reverse quaternions?

Inverting or conjugating a rotation quaternion has the effect of reversing the axis of rotation, which modifies it to rotate in the opposite direction from the original. That is, if a point is rotated to a new position using q, then rotating it again using q−1 or q* will return it to its original location.

How do you multiply a vector by quaternion?

As for the multiplication with a vector, you just extend the vector to a quaternion by setting a quat’s real component to zero and its ijk components to the vector’s xyz. Then you do the quaternion multiplications to get v’, and then extract the ijk components again.