Re: vector components and notation

["RAUBER, JOEL" <JOEL_RAUBER@SDSTATE.EDU>]

A trend that I think I have noticed lately (and which matches how I present
the material for the last several years) is to distinguish between
"components" and "component vectors".

The unfortunate part is the close similarity of names.

"component vector" = those vectors that are parallel to coordinate axes that
add to the vector of interest.

"component" = as Bob says below, the projection of the vector of interest
onto the axes, what is traditionally labeled A_x and A_y (signed scalar
quantities).

Joel

> -----Original Message-----
> From: Bob LaMontagne [mailto:rlamont@POSTOFFICE.PROVIDENCE.EDU]
> Sent: Monday, September 16, 2002 5:14 PM
> To: PHYS-L@lists.nau.edu
> Subject: Re: vector components and notation
>
>
> One of the reasons that I have used Serway (Physics for
> Scientists and Engineers) is because I feel he is one of the
> few authors who really presents the concept of components
> correctly and also very clearly explains the concept to the
> student. The Vectors Ax and Ay that can add vectorially to
> form the vector A are _not_ the components. The true
> components are the _scalar_  projections of the the vector A
> onto the x and y axes. As scalars, they can be positive or
> negative depending on which side of the axes they project.
> Serway, to his
> credit, carefully avoids assigning a sign to the vectors Ax
> and Ay. I will certainly take a good look at Hecht's 2003
> text if it also takes a similarly clear and correct approach.
>  Hecht is extremely skilled in getting across the
> 'conceptual' aspect of physics.
>
> Thanks - Bob at PC
>
>
> Joe Heafner wrote:
>
> > Fellow readers,
> >
> > 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.