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

[Fido]

Homework Statement

Show that for any scalar field α and vector field B:

∇ x (αB) = ∇α x B + α∇ x B

Homework Equations

(∇ x B)_i = ε_ijk v_k,j
(∇α)_i = α_i
(u x v)_i = e_ijku_jv_k

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:
ε_ijkB_k,j α_i

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:
α_i x B + α∇ x B

 

[Simon Bridge]

Nabla cross product with something is the curl of the thing right?
http://www.math.ucla.edu/~ronmiech/Calculus_Problems/32B/chap14/section5/930d31/930_31.html

 

[Fido]

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

 

[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.