inner product and parallelogram equation

Question

So I know that if ∥⃗v∥ is associated to an inner product •, then it satisfies the parallelogram equality:

∥⃗u + ⃗v∥^2 + ∥⃗u − ⃗v∥^2 = 2(∥⃗u∥^2 + ∥⃗v∥^2)

for all ⃗u and ⃗v.

I also know that I need to find two vectors for which the parallelogram equality fails.

For example I take ⃗u = ⃗e1 and ⃗v = ⃗e2.

After this step I am not really sure what to do.

Answer 1

If $u = (1,0)$ and $v = (0,1)$, we have $2(\|u\|^2 + \|v\|^2) = 4$ but $\|u + v\|^2 + \|u - v\|^2 = 8$. Therefore the parallelogram law does not hold.