How do I cross Del with (scalar*vector)?

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1. Sep 5, 2016

Fido

1. The problem statement, all variables and given/known data
Show that for any scalar field α and vector field B:

∇ x (αB) = ∇α x B + α∇ x B

2. Relevant equations

(∇ x B)i = εijk vk,j
(∇α)i = αi
(u x v)i = eijkujvk
3. The attempt at a solution
Since α is a scalar i wasn't quite sure how to cross it with ∇

So on the left side I have:
εijkBk,j αi

I'm pretty sure I'm supposed to solve out the left side to get to the right because that what I did for the previous problem.

The right side all I could get it to simplify to was:
αi x B + α∇ x B

2. Sep 5, 2016

3. Sep 5, 2016

Fido

So does that mean that my left side is just εijk Bk,j α ? I feel like it's supposed to be broken up into two parts

Last edited: Sep 5, 2016
4. Sep 5, 2016

Simon Bridge

It needs to if you are going to get RHS = LHS ... if in doubt go back to the definitions.
Maybe look at $(\alpha B)_i$ also see what the RHS looks like in components. Play around until you understand it.