Magnitude and area of a parallelogram

From: Cassandra Thompson (newsgroups_at_calebsoftware.com)
Date: 12/04/04 

Date: Sat, 04 Dec 2004 23:41:36 GMT

Firstly, I am not sure if it annoying that I am using the same thread to
ask so many question, however I thought it might be equally annoying to
be starting lots of new threads. At least if it is in the one thread you
have a good idea of my background.

This question is not part of the assignment, however working through the
assignment has shed light on a couple a few equations, which in turn
have confused me. I am hoping to ask some questions.

We have two vectors:
a = (1, -2, 1)
b = (3, 1, 0)

The cross-product of these two vectors is:
a x b = (-1, 3, 7)
let w = (-1, 3, 7)

Okay all good so far I think. Now if I want to find the magnitude of w,
I treat it exactly the same as I would if I wanted to find the magnitude
of a and b.
|w| = sqrt((-1)^2 + 3^2 + 7^2) = sqrt(17)

However I also know that the following formulae is true:
|a x b| = |a||b| sin&

Is the |a x b| in this formulae the same as |w|?
Does |w| = |a||b| sin & hold true?
Does |a x b| = sqrt(17)?

I guess I am just confused by the to different formulaes.
ie
|w| = sqrt(w_1^2 + w_2^2 + w_3^2)
|a x b| = |a||b| sin&

Thanks,
Cassandra.

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