In the preface to his new alg/trig based textbook (Physics: Algebra/Trig, 3rd edition. Brooks/Cole, 2003), Hecht finally begins to address the issue of contradictory notation used to explain vector components. Components are two orthogonal vectors that add to give a given original vector. Vectors don't have an algebraic sign about them; they are independent geometric entities with their own properties. Vector *components*, however may be treated as positive or negative with respect to a given coordinate axis. Herein lies the confusion.
How can a vector *component* be both a vector and a scalar? I don't think that's possible. When we form the dot product of a vector with, say, an arbitrarily defined x-axis we get a *scalar*. How can we then justify calling this thing a vector? I don't think we can call it the magnitude of a vector either since magnitudes are always algebraically positive. Dot products can be positive or negative. I suspect the problem lies with the flawed usage of the term *component*.
I am currently reading, for the second time, Gabriel Weinreich's excellent book Geometrical Vectors. If anyone would like to discuss certain aspects of this excellent little book on this list, I'd be grateful becuase I have some questions about some things.
Cheers,
Joe Heafner - Instructional Astronomy and Physics
Home Page http://users.vnet.net/heafnerj/index.html
I don't have a Lexus, but I do have a Mac. Same thing.