Stimulated by the unit vector thread, I was reading John Denker's discussion
at http://www.av8n.com/physics/intro-vector.htm and would like to question
John (and others) about the dot product definition.
I agree with and appreciate John's comments that the dot product of vectors
A and B should be defined as ABcos(theta). What I question is John's
comment that A dot B should not appear in physics textbooks as AxBx + AyBy +
AzBz, unless what he specifically means by "not appear" is that it should
not appear as a definition. I think it should appear as one way to evaluate
A dot B, and indeed can then be used along with the definition of A dot B to
find the angle between A and B.
When I am discussing the dot product in my physics class defined as
ABcos(theta), the question inevitably crops up "how do we find theta?" This
is especially a common question in three-dimension situations.
It seems to me that showing there is a way to compute the dot product by
choosing a basis, expressing the vectors using that basis, then applying the
AxBx + AyBy + AzBz procedure is a worthwhile thing to teach and can
appropriately appear in physics books. At the same time we can also discuss
why ABcos(theta) is a good definition but AxBx + AyBy + AzBz is really a
"computational procedure" (rather than a definition).
Comments?
Michael D. Edmiston, Ph.D.
Professor of Chemistry and Physics
Bluffton University
1 University Drive
Bluffton, OH 45817
419.358.3270
edmiston@bluffton.edu