6 ms·
ij=k means that the product of two of the three unit vectors must equal the third, which is a fundamental, non-arbitrary property, and without loss of generalit
by HelloNurse 7y ago
ij=k means that the product of two of the three unit vectors must equal the third, which is a fundamental, non-arbitrary property, and without loss of generality the signs can be chosen so that ij=k (rather than ij=-k).
Adding ii=-1,jj=-1,kk=-1 it's easy to deduce ijk=-1, j=ki, and i=jk, and ij=-ji, ik=-ki, jk=-kj.
Having to remember, out of the six possible permutations, that one of the three correct formulas is the one with the symbols in alphabetical order is much better than remembering the "right hand rule" (or was it "left hand rule"?).
- Certhas 7y agoBut 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.