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But why would you remember a right hand or left hand rule? bivectors don't require you to. x ^ y = x y - y x^T is the generator of the rotation that rotates a
by Certhas 7y ago
But why would you remember a right hand or left hand rule? bivectors don't require you to.
x ^ y = x y - y x^T is the generator of the rotation that rotates a vector in the direction y into the direction of x. There is absolutely nothing to remember, and you can work this out immediately from just multiplying out the matrices:
exp(epsilon x ^ y) v ~ v + epsilon x ^ y * v
= v + epsilon x (y,v) - epsilon y (x,v)
We add a little bit in the x direction and remove a bit in the y direction.
The whole view of rotations happening round an axis is just a coincidence of 3d space, it doesn't make sense in higher dimensions. On the other hand rotations always do (compose into ones that) happen on planes.
- HelloNurse 7y agoThat's my point, you need to remember "ijk" in order (i.e. i=jk or ij=k) in a purely algebraic definition instead of a meaningless and complicated "handedness" rule involving rotations.
- papln 7y agoIn my college in Antarctica we were taught x ^ y = y x^T - x y exp(ε x ^ y) v ~ v + ε x ^ y * v = ε y (x,v) - v + ε x (y,v) because we use the left-hand rule down under.