[gameprogrammer] Re: Meaning of cross product

If by 'trapezoid' you really mean 'parallelogram' you are right and if
not then your theory is wrong.  The trapezoid may or may not have
parallel sides.  A parallelogram is a special case of trapezoid which
has opposite sides parallel.  The property of parallelism makes the
'magnitude' of the cross product equal to the area of the parallelogram
as follows:
Assume two vectors A and B connected at the tail, i.e., adjacent edges
of a parallelogram.  A special case of the two vectors is when the angle
between the two vectors is 90 degrees, which forms a rectangle and the
special case of the rectangle is the square (magnitude A is the same as
magnitude B).
For notational purposes, say the magnitude of B is written |B| and the
magnitude of A is written |A|.
The area of a rectangle is height times length.  Restated the area of
the rectangle is the magnitude of vector A, |A|, times the magnitude of
vector B, |B|.  Restated Area of rectangle (or square) is |A||B|.
Now suppose that the angle, theta, between the two vectors, A and B, is
not 90 degrees.  Well, then the height of the resulting parallelogram is
the magnitude of the one vector, say B, times the sin of the angle,
theta, height = |B|sin(theta).  In other words, the area is now the
height times the magnitude of the other side, vector A,
|A||B|sin(theta).
Now, remember the definition of the magnitude of the cross product |AxB|
is defined as |A||B|sin(theta). (This is not really a proof.)
In the case of the trapezoid, at least two sides are not parallel. In
other words opposite angles (or sides) must differ by some amount. If
the angle difference is small than the error in using the cross product
as an approximation to the area is small. With a little bit of
visualization you will be able to see that the greater the difference in
angle means the greater the error in using the cross product as an
approximation to the area of a trapezoid.
 
>  -----Original Message-----
>  From: gameprogrammer-bounce@xxxxxxxxxxxxx [mailto:gameprogrammer-
>  bounce@xxxxxxxxxxxxx] On Behalf Of Kevin Jenkins
>  Sent: Wednesday, October 13, 2004 3:39 PM
>  To: gameprogrammer@xxxxxxxxxxxxx
>  Subject: [gameprogrammer] Meaning of cross product
>  
>  I have a theory that the magnitude of the cross product of two
position
>  vectors gives you the area of the trapezoid those vectors form,
assuming
>  those two vectors were two adjacent edges of the trapezoid.
>  
>  It's too much trouble to do the proof so I wanted to ask and see if
>  anyone knew this for sure?
>  
>  
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