Given a three-axes coordinate system ${1,2,3} $ by the right-hand rule, and a new coordinate system ${1',2',3'}$ , I know that one can define a vector $\vec{x}$ to be something that obeys the following transformation rule:
If $(x_1 , x_2 ,x_3 )$ are its locations in the system 1,2,3 , and $(x_1 ' ,x_2 ' ,x_3 ' ) $ are its locations in the system 1',2',3' then: $x_i = \ell _{ij} x_j ' $ (I used here Einstein's summation convention) [where $\ell_{ij}$ is the angle between old axis i and new axis j' ] .
I never understood this property. Can someone please explain to me how can I derive this transformation ? (i.e.- how can I show geometrically that a vector in $\mathbb{R}^3 $ is indeed a vector)
In addition, how can one prove the identity $\ell_{ij} \ell_{ik} = \delta _{jk} $ without using the identity I mentioned above ?
Thanks in advance