# Cross product

• I
• prashant singh
In summary, the cross product is defined as A x B = |A||B|sin(theta) where theta is the angle between A and B. The matrix formula, also known as the cross product determinant, is used as a mnemonic device to compute the cross product in a Cartesian coordinate system. However, it is not the definition of the cross product, which can be geometrically defined as A x B = (|A||B| sin(θ)) u, where u is a unit vector perpendicular to both A and B, determined using the right-hand rule. Some tutorials and videos may use the matrix formula to prove that ||A x B|| = ||A|| ||B|| sin(theta), but this is just a manipulation of

#### 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

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
What do you mean by "matrixx formula"?

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

jedishrfu said:
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|.
jedishrfu said:
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.

• prashant singh