On Friday 05 August 2011 20:22:21 BGB wrote: > On 8/5/2011 11:56 AM, Wesley Smith wrote: > >> vectors are nice though. > >> for example, in the book I had, some aspects of the topic were expressed > >> in terms of a mess of trigonometry which wouldn't really work correctly > >> in 3D. some of these topics were fairly simple/elegant-looking if > >> expressed with vectors. > >> > >> so, linear systems and vectors, probably could do fairly well I think. > >> > >> more so, linear systems and vectors would give students a model that > >> they could more easily use with or test on a computer. > > > > The problem with vectors is that they are closed only under > > subtraction, addition, and scalar multiplication. As soon as you take > > the product of two vectors, you get a completely different > > object/type, namely quaternions in the 3D case and rotors in the > > general case. Vector algebra's widespread use is an artifact of the > > vector v. quaternion debate from the 19th century. It's not the most > > versatile tool and hides lots of structure and symmetries. Similar to > > the "gotos considered harmful" mantra, I would add "vector algebra > > considered harmful". > > typically, vector multiplication is treated as either dot-product or > cross-product (with cross-product only existing in certain numbers of > dimensions, such as 3 and 7, and "sort of" in 2). > > for 2D and 3D, vector math works fine. > > and in 3D, vector math is much nicer than 3D trigonometry. > > > quaternions also exist, and can be useful, but for many practical tasks > (such as performing rotations) share an ongoing battle with matrix-math, > and in some cases with angle-based systems. > > may as well include quaternions as well though, since as I see it there > is no fundamental conflict between them and vectors.
The irony, like many "competing theories", being that they're really just the same thing, when looked at in the appropriate way ;) http://blog.sigfpe.com/2006/08/geometric-algebra-for-free_30.html Cheers, Chris _______________________________________________ fonc mailing list [email protected] http://vpri.org/mailman/listinfo/fonc
