Determinates to Calculate Cross Product

[QuarkCharmer]

Homework Statement

Not really a homework problem, just a general question.

Homework Equations

Sorry, I don't yet know how to create arrays/matrices in latex (this is a gif)

The Attempt at a Solution

In the above image, why is $c_{1}$ positive, $c_{2}$ negative, and $c_{3}$ positive?

Aside from that, I think I understand the rest. I would assume c from the formula to be the unit vectors denoting direction, A would be the first vector in the product, and B would be the second. From there,

$c_{1}$ times $(a_{2}b_{3} - b_{2}a_{3})$

would be the $c_{1}$ component of the product and so on...

But why would the y/j component be negative in the formula?

 

[LCKurtz]

Look up expansion of determinants by minors:

http://mathworld.wolfram.com/DeterminantExpansionbyMinors.html

 

[QuarkCharmer]

Ahh, I see. So it's going to alternate every time as the exponent (i+j) becomes odd/even. I can see how this would get really tedious for matrices larger than 3x3.

Thanks for the information!

 

[flyingpig]

It's just the nature of it isn't it? It's like looking at the mirror