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.
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