Confused with Right hand in context of Cross Products

[screamingman]

This is not a homework problem, but pertains to the class (Calculus II) that I am taking.

I am very confused about how to think intuitively about determining orientation of a cross product.

My understanding of the right hand rule in determining the orientation of the third axis is that by convention, if two axis are oriented positively in a certain way, then by convention the third should be pointed a specific way, and you use the right hand rule to determine this way. This makes sense to me.

But when it comes to the cross product, I am so confused by the intuition behind determining the orientation of the vector. The textbook merely says that if you curl your fingers from a to b the direction that your thumb is pointed determines the direction of the cross product. Why? I have searched this question on the internet and on physics forums and still do not understand. I have seen responses that it is also a convention, but I guess I don't understand the motive for the convention.

I do not have any physics background (well besides a intro mechanics class) that has used the right hand rule, so that may be part of the problem.

The closest answer I saw to my question was here: http://www.scienceforums.net/topic/74133-what-is-the-intuition-behind-the-right-hand-rule/ ,but I am still confused.

I guess my understanding right now is that when you cross two vectors, you get a vector perpendicular to that vector and because that vector could be oriented in either direction, we use the right hand rule to decide a direction so that we have a convention by which we can agree it is oriented? Why should ixk be oppisitelly oriented to jxi?

Thank you in advance.

 

[jedishrfu]

Your understanding is very close. The cross product definition says A X B is a vector of length |A||B|sin (theta) where theta is the angle between them which is perpendicular to both A and B. In 3D space then there are two vectors that are perpendicular to both A and B and so the righthand rule conventions chooses one of the two perpendicular vectors and says it is in the positive direction.

This convention is used throughout vector algebra and is reflected in the XYZ coordinate system scheme of finding the A X B resultant vector:

|  i  j  k |
| Ax Ay Az |    =  (AyBz - AzBy) i  +  (AxBz - AzBx) j  +  (AxBy - AyBx) k  = a vector that is perpendicular to both A and B
| Bx By Bz |

You can use the vector dot product to show that the resultant vector is truly perpendicular to A and the B.

http://en.wikipedia.org/wiki/Vector_cross_product

 

[screamingman]

Thanks! So we have chosen that perpendicular vector found by the cross product is always positive? I still don't understand why ixj is oppositely oriented to jxi.

 

[vela]

I'm not sure what kind of answer you're looking for here. Just apply the right-hand rule to both products, and you'll see you get different answers. If you're thinking they should be the same because the order shouldn't matter, that's wrong. The order does matter. The cross product isn't the same as the multiplication of regular numbers. If you look at the determinant jedishrfu wrote, swapping the vectors A and B means exchanging two rows of the matrix, which means the determinant changes sign.

 

[screamingman]

Never mind, I get it now, a bit more and I think to reach higher level of understanding in this matter, I will have to understand its application to physics.