Why is the cross product orthogonal to the plane spanned by two vectors?

[zeion]

Homework Statement

Hello. I am relatively new to this subject please forgive my incompetence.
Please correct me if I have misunderstandings.

I understand that the cross product of two vectors (say A and B) in R3 is a vector that is orthogonal to both A and B. But how A x B be orthogonal to the plane spanned by A and B?

I don't understand what this plane spanned by A and B should look like geometrically.

Wouldn't A x B be parallel to the plane spanned by A and B?

 

[LCKurtz]

Think of your two vectors A and B positioned with their tails together and assume they are not parallel. Think of the parallelogram created using those vectors as two sides of the parallelogram. That parallelogram is part of the plane spanned by A and B. A x B is perpendicular to that plane.

 

[zeion]

Oh okay I got that mixed up then. Thanks.

So then the plane by spanning A and B has the vectors A and B as the boundaries right?
Why does it make a parallelogram when there is only two vectors?

Now my next question is how to understand the weird order in which you multiply the elements of A and B to get A x B?

The vector gotten from A x B is only from the point where A and B intersect?

 

[diazona]

A plane has no boundaries. The vectors are two sides of the parallelogram, and the other two sides can be drawn parallel to the original two vectors to complete the parallelogram.

Wikipedia has some nice pictures which may help clarify things:
http://en.wikipedia.org/wiki/Cross_product