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I just had a thought (which probably isn't original and it is evolvingas I write this) but how about
putting the sign with the unit vector rather than the coefficient outfront. So a motion in the -x
direction with a speed of 2 m/s would bequantity. The coefficient is
v = (2 m/s)(-i^)
This emphasizes the vector nature AND the magnitude being a positive
always positive and carries the units, while the vector nature iswrapped up into a dimensionless,
unit length vector.(|A|)(A^), which matches
A speed of 2 m/s 45 degrees below the x axis could be either
v = (1.414 m/s)(i^) + (1.414 m/s)(-j^)
or
v = (2 m/s)(0.707 i^ - 0.707 j^)
The first form is easier for addition; the second emphasizes the speed.
In general then, any vector would be written in "standard form" as A =
exactly the mantra of "a vector is a quantity with a magnitude and adirection". The "quantity"
contains the magnitude and units, while the "direction" is always givenby a unit vector, which you
can either leave as A^ or convert to (cos(theta) i^ + sin(theta) j^ )."parts" of a vector. In
I would advocate always putting in the parentheses to emphasize the two
fact, other than this emphasis, it's of course no different thanstandard polar or cartesian
coordinates.
Comments???
Tim Folkerts