Why is the Vector (Cross) Product pxq Perpendicular to a Plane?

[matqkks]

Given two vectors p and q, why is the vector (cross) product pxq perpendicular to the plane containing these vectors?
Is there a geometric or physical way of explaining why?
Are there any real life examples we can draw from?

 

[Landau]

Well, what is your definition of cross product? Sometimes this (perpendicularity) is part of the definition.

 

[Rasalhague]

Two links posted recently in these forums are online introductions to multivariable calculus which each contain a chapter relating the algebraic expression for the components of a cross product to the property of being perpendicular to its factors:

http://synechism.org/drupal/cfsv/
http://www.owlnet.rice.edu/~fjones/

Greg Egan used to have a nice demonstration of the correspondence between geometric and componentwise definitions of the dot product (which is taken as the starting point by the links above), but I can't seem to find it now. Never mind, Google knows lots more. You can shop around for the one that makes most sense to you.