If I have a vector w and index it with, say, j then if w appears in another term in the equation do I use the same index i.e. j? Thanks
An example would be helpful. Usually $\displaystyle w_j$ is considered a function that maps a natural number $\displaystyle j$ into the $\displaystyle j$th component of $\displaystyle w$ (or maps a sequence of vectors $\displaystyle w$ into the $\displaystyle j$th vector, depending on the meaning of "index $\displaystyle w$ with $\displaystyle j$"). Then, as in any function, the argument can be anything, e.g., 0, $\displaystyle j + 1$, $\displaystyle w_{i}$, etc. But if you make a convention (even implicitly) that $\displaystyle j$ ranges over vector indices, it is better to keep using $\displaystyle j$ for this and not change it to, say, $\displaystyle i$ without reason.
Thank you.
$\displaystyle (a\times b)/times(a\times c)$
= $\displaystyle (\epsilon_{ijk}a_{i}b_{j}\hat{e_k}) x (epsilon_{ilm}a_{i}c_{l}\hat{e_m})$
=$\displaystyle ( \delta_{jl}\delta_{km}- \delta_{jm}\delta_{lk})\epsilon_{kmn}\hat{e_n}a_{i }a_{i}b_{j}c_{l}$
I appear to have gone wrong