[Phys-l] vector basics

[John Denker <jsd@av8n.com>]

On 03/31/2009 09:21 AM, Bill Nettles wrote:
>  ... say, place the origin at the intersection of the
> "y"-line of the CM of 2 and the velocity vector through the CM of 1,


Velocity vectors don't go "through" anything, ever.

Vectors have direction and magnitude.
That does not mean direction, magnitude, and point-of-attachment.
It means direction and magnitude, period.

The following three diagrams all represent the *same* vector:

      ----->

                                  ----->
                   ----->


This may seem like a small point, but sometimes small points are
important.  It might be argued that "velocity vector through such
and such point" is a convenient shorthand, but I don't buy it.  We 
routinely insist that students respect the distinction between mass 
and weight, the distinction between force and pressure, the distinction 
between mean and standard deviation, etc. etc. etc.... and IMHO it 
is at least as important to keep straight the distinction between the 
velocity and the position of the CM.  (In the recent angular momentum 
problem it suffices to make the distinction between the velocity and 
the lever arm.)

The velocity and the position of the CM are two different vectors.  
There is an important distinction between one vector and two vectors.

If you want to get fancy you can introduce the bivector r/\p, which is
a first-class geometric object unto itself, but still there is an important 
distinction between a vector and a bivector.