Question

In my course of special relativity we are introducing tensors: however, before doing that, my professor sort of re-defined vectors saying that in a 3D euclidean space, A

can be called a vector if, under a rotation, it transforms like the coordinates. So, if
x′i=R_ijx_j
where R
is the rotation matrix and it satisfies R^TR=RR^T=1
(I'm calling 1
the identity matrix), then A
is a vector if
A′_i=R_ijA_j
Now, he told us to try to prove that, if A
and B
are vectors, then the cross product V=A×B
is a vector too.

I know I should get the following:
V′_i=R_ijV_j=R_ij(ε_jlmA_lB_m)
My professor also told us that the following identity is true:
ε_abcR_ajR_bkR_cm=det(R)εjkm
This is what I've done:
V′_i=(ε_ijkA_jB_k)′=εijkA′_jB′_k=εijk(R_jlA_l)(R_kmB_m)
I have no idea how I should apply the identity my professor gave us and I don't know where that third matrix comes from.

Answer 1

The cross product of two vectors is also a vector.

Step-by-step explanation:

To prove that the cross product of two vectors is also a vector, we start with the definition of the cross product: V = A x B. If A and B are vectors, then we can represent them as A = (Ax, Ay, Az) and B = (Bx, By, Bz). Using the cross product formula, we get V = (AyBz - AzBy)i + (AzBx - AxBz)j + (AxBz - AyBx)k.

Since V is expressed in terms of the unit vectors i, j, and k, which are vectors themselves, the cross product V is also a vector. Therefore, the cross product of two vectors A and B is indeed a vector.