Is There Proof for the Cross Product Matrix Formula?

[prashant singh]

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

 

[fresh_42]

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

What do you mean by "matrixx formula"?

 

[jedishrfu]

The matrix formula aka the cross product determinant is more of a mnemonic device to help you compute the cross product in a Cartesian coordinate system.

from wikipedia (see the matrix notation and the Sarrus' Rule:

https://en.wikipedia.org/wiki/Cross_product

 

[prashant singh]

See jedishrfu answer

 

[jedishrfu]

Vector cross product is defined as AxB = |A||B|sin(theta) where theta is the angle between A and B.

Here's a small tutorial on it:

http://users.math.msu.edu/users/gnagy/teaching/11-fall/mth234/l04-234.pdf

 

[Mark44]

Vector cross product is defined as AxB = |A||B|sin(theta) where theta is the angle between A and B.

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|.

Here's a small tutorial on it:

http://users.math.msu.edu/users/gnagy/teaching/11-fall/mth234/l04-234.pdf

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]

My apologies I forgot to mention that the resultant vector is perpendicular to both A and B and follows the right-hand rule for direction as determined via rotation from A to B.

Alas, still learning what I forgot and forgetting what I've I've relearned.