The discussion revolves around the proof of the matrix formula for the cross product, particularly its relationship to the magnitude of the cross product and its geometric interpretation. Participants explore various aspects of the formula, including its use as a computational tool and its definitions.
Participants do not reach a consensus on the proof of the matrix formula or its definitions, with multiple competing views and interpretations presented throughout the discussion.
Some participants note the lack of geometric definitions in certain resources and the potential confusion surrounding the distinction between the magnitude of the cross product and its vector form.
What do you mean by "matrixx formula"?prashant singh said:Is there any proof for the matrixx formula of the cross product. I am asking this because I have seen many videos and they have used the matrixx formula and then proved that ||A X B|| = ||A|||B||sin(theta), khan academy also used the same method
Sorry, this isn't the definition of the cross product A x B. It's the definition of the magnitude or norm of A x B; i.e., |A x B|.jedishrfu said:Vector cross product is defined as AxB = |A||B|sin(theta) where theta is the angle between A and B.
In the pdf in the link, the author never gives a geometric definition of the cross product. This product could be defined as A x B = (|A||B| sin(θ)) u, where u is a unit vector that is perpendicular to both A and B. The orientation of u can be determined using the right-hand rule (i.e., if you let the fingers of your right hand curl around from A to B, your thumb will point in the direction of u).jedishrfu said:Here's a small tutorial on it:
http://users.math.msu.edu/users/gnagy/teaching/11-fall/mth234/l04-234.pdf