Cross product in matrix determinant form

[Nick R]

Everything I have read indicates that the cross product is simply defined as

a x b = i( ay*bz - az*by) - j( ax*bz - az*bx ) + k( ax*by - ay*bx )

and that it just so happens that there is a shorthand notation of cross product in matrix determinant form.

How is the cross product formulated? Is it worked out geometrically, or does it reflect some property of the determinant? Or either?

The most revealing thing I have run into so far is that on mathworld it says the determinant of a square matrix has the interpretation of the "content of the parallel piped spanning the column vectors". As I recall the magnitude of the cross product is equal to the area of the parallogram associated with the 2 vectors so there must be some sort of link here.

 

[ForMyThunder]

The cross product a×b can also be defined as

a×b=|a||b|(sin θ)n

where n is the unit normal to the plane of a and b and θ is the angle

between a and b.

a×b is a vector that is normal to the plane of a and b with a

magnitude equal to the area of a paralellogram with sides a and b.

 

[quasar987]

I assume that the shorthand notation of cross product in matrix determinant form you are talking about is this thing: http://en.wikipedia.org/wiki/Cross_product#Matrix_notation.

But that is only a notation, useful to remember how the cross product is computed, because the entries of that matrix are vectors on the first row, which does not agree with the definition of the determinant as a function from matrices to real numbers.

There is another characterisation of cross product that we can use to generalize the product to higher dimensions. It is that u x v is the unique vector such that <u x v,w> = volume of parallelepiped spanned by u, v and w = det(u v w). (existence and uniqueness of such a vector is guarented by the Riesz representation theorem)