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Re: [Phys-l] Motion in 1D, vectors and vector components



On 08/15/2007 10:45 AM, Jeffrey Schnick wrote:

I think it would be beneficial to physics learners in general if we
could clear up this confusion once and for all. Consider the following
scenario: A block has a bunch of forces, all lying in one plane, acting
on it, including a force exerted on the block by a rigid strut (of
negligible mass) which lies in the plane of the forces and is
pin-connected (frictionless pins perpendicular to the plane of the
forces) on one end to the block and on the other end to something else.
Other than the pins, nothing is exerting a force on the strut. The
orientation of the strut is specified in the problem statement. The
question is, what is the force exerted on the block (plus the pin on the
block, which is considered to be part of the block) by the strut?
Enough information about the other forces is provided to make the
problem solvable. Each student that solves the problem draws a free
body diagram using a point of view in which the strut extends up and to
the right at the specified angle above the horizontal.

Pin-connected? That's amusing. There's some real-world relevance
there. I'll bet it takes the students a while to understand the
ramifications of pin-connectedness.

[ whole passel of student misconceptions -- snipped ]
[ see https://carnot.physics.buffalo.edu/archives/2007/8_2007/msg00219.html ]

Now it is time for the teacher to prepare a mini lecture for the entire
class. If you were the teacher, besides the obvious statement about the
magnitude of a vector being positive by definition, what would you say
to the students?

Four suggestions, taken from the newly-revised
http://www.av8n.com/physics/intro-vector.htm#sec-suggestions

1. If you are diagramming vectors, draw the physically-relevant
geometrical vectors. Draw them to scale. This is a purely analog representation.
Such vectors can be compared visually, comparing them as to direction and
magnitude. Such vectors can be added graphically, adding them tip-to-tail.

2. If you are going to quantify the vectors in terms of matrix
elements, indicate what the basis vectors are. The matrix elements don’t
mean anything if we don’t know the basis vectors. The connection between
matrix elements, basis vectors, and general geometric vectors is defined
by
http://www.av8n.com/physics/intro-vector.htm#eq-expand-moe

3. Combining the two previous items, it is likely that you will want
to draw both the physically-relevant vectors and the basis vectors, as in
http://www.av8n.com/physics/intro-vector.htm#fig-vector-vs-basis
Make it clear which is which.

4. It is not worth using distinctive symbols for vectors (such as
boldface letters, or letters decorated with arrows over them). In the
expression F=ma, even without decorations it is well known that F and a
are vectors, while m is a scalar. By way of analogy: we use ordinary letters
of the alphabet for lengths, areas, volumes, et cetera, and we are able to
tell which is which from the definition of the symbols. We can analyze the
dimensions and units to check that we have done everything right. By the
same token, we can use ordinary letters to represent vectors. The vectorial
character is part of the “units” and “dimensions” of the quantity.

The convention of using boldface to represent vectors fails both in
handwritten notes and in ascii email. The convention of drawing an arrow
atop the symbol fails in email. All such decoration conventions fail
miserably in the context of Clifford algebra, aka geometric algebra, where
some quantities have both a scalar piece and a vector piece.

The convention of using a decorated letter to represent a vector while
the corresponding undecorated letter represents the magnitude of the vector
is cute, but is not worth the trouble. If you want the magnitude of F, write
|F| explicitly. The cost of writing |F| when you want the magnitude is
infinitesimal compared to the cost of decorating F when you want the whole
vector.

If you choose to designate the strut-force-related arrow as a basis
vector, how would you indicate on the diagram that it is a basis vector
rather than a physical vector?

I might call it e1 on the diagram, and indicate in the legend
that e1 e2 e3 are basis vectors.

Label it "basis vector"? (Does one then
also need to label each physical vector "physical vector"?)

That seems unnecessarily verbose. It might perhaps be justifiable
in special situations.

Or would
you use two parallel lines ==== to depicted the shaft of a basis arrow
vector rather than the single line ---- that you use to depict the shaft
of a physical vector? Where would you put the "F_s"?

Draw the physically-significant vector separately from the basis vector.
Label the physically-significant vector F_s.

========

BTW I think drawing a unit vector in the same direction as the
strut is a cop-out, i.e. a way of not answering the question,
because it begs the question of the direction of the force.
Saying the force is in the direction of the strut isn't a
sufficiently quantitative answer to the assigned question,
unless they quantify the direction of the strut.
If they're going to quantify the magnitude of the force (15N)
they should quantify the direction with comparable precision.

============================

Let me answer the question that was explicitly un-asked. I
think nearly all the important content here involves figuring
out what is a magnitude, what is a matrix element, what is a basis
vector, and what is the vector of interest. I consider this to be
an extension of the idea of unit-analysis and dimensional analysis;
the vectorial character of a thing is part of the dimensions and/or
units of the thing.

a) Writing |Fs| = -15N is absurd, because the magnitude is positive.
b) Writing Fs = -15N is absurd, since the LHS is a vector while the
RHS is not.
c) Writing Fs = 15N e_x might be OK.
d) Writing Fs · e_x = -15N might be OK.
e) Note that (c) and (d) are not generally synonymous.