I'll state a preference for indicating vector quantities with an arrow
over the symbol in the introductory course.
a) it reinforces the distinction between vectors and numbers; lack of
which is often at the root of many beginning student mistakes and
misunderstandings
b) in the context of the terminology discussion
how is one to distinguish the array element v_x
from the projection, i.e. component vector, v_x
its easy and self-describing to use the venerable arrow over the symbol
for the projection.
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In introductory physics I certainly don't want to use
<e_x|v> the inner product, in lieu of v_x
or |e_x><e_x|v> in lieu of v_x with an arrow over it. Or similar such
notations.
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On another note: The decision of what to use is not done in a vacuum,
the students often have a proscribed text that you are using and you
don't want to stray too far from what the texts terminology and notation
without very good reason.
Most texts either use an arrow over the symbol or use a bold face font.
I eschew the bold face font as I can't write that way on the board, and
using blackboard bold fonts is tiresome and more time consuming IMO. If
the text uses arrows, then you are doing the same. If text is using
bold-face, it is easy to justify to your students your use of arrows on
the blackboard.
This brings up the terminology issue again. FWIW (for what its worth)
Some data from widely used books or books I like for various reasons or
books within arms reach.
Halliday, Resnick, Walker: Use the term projection, but in diagrams a_x
the projection of a is indicated by an arrow; i.e. a vector. In all
formulas a_x is the array element. And these are both amigously
referred to as components and hence projection can mean either.
Knight: uses "component vector" and "component"
Halliday, Resnick and Krane: use "vector component" and "component"
Crummett and Western: use "component" for the array element and don't
provide a name for |e_x><e_x|v_x>
Lea and Burke: use "component vector" and "component"
Young and Freedman: use "component vector" and "component"
Hecht: use "component vector" and "scalar component". He also simply
uses "component" for the latter.
Moore: uses "component vector and "component"
Chabay and Sherwood": use "component" for the array element but don't
provide a name for the projection, though they do use constructions in
formulae like
v_vec = v_x i_hat + v_y j_hat + v_z k_hat
usw. (und so weiter) etc.
This unscientific sampling seems to indicate that authors that care to
make the distinction and provide names use
"component vector" or its reverse and "component"
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Joel Rauber
Department of Physics - SDSU