Direct Proof of Div, Grad, and Curl Operator Identities

[unscientific]

Homework Statement

Homework Equations

The Attempt at a Solution

For part (c), I showed that the tripple cross product = 2a using einstein notation. Then, I showed that 2∇(a.r) = 2a which is the same as LHS. I don't think this is as elegant as it can get..

How do I prove it directly? I've looked up some operator identities:

 

[BruceW]

Your method is fine. I would have done the same thing. I know what you mean though. It feels like there should be some kind of shortcut method.