Vector Calculus - Tensor Identity Problem

[mille2eo]

Homework Statement

Homework Equations

The Attempt at a Solution

I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus.

This is my attempt at the cross product and dot product written out. I don't know how to proceed to make the left hand side equal the right hand side.
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[eys_physics]

Use the expression for the scalar product between two vectors $\mathbf{a}$ and $\mathbf{b}$:
$$\mathbf{a}\cdot\mathbf{b}=\sum_i a_i b_i $$