Demonstrating the norm of a sum of vectors.

Question

I need help in this question, it's a true or false question, where it asks to prove or a counter proof of several items. There's one item in particular where I can't understand. If it's true, what properties or what I must use to prove.

Question: $\|u+v\|^2 = \|v\|^2 + 2 \|v\| \|u\| + \|u\|^2$,
where every $u$ and $v$ are vectors.

Just like this: https://i.stack.imgur.com/s7KaW.png

Answer 1

It is clearly false when $v=-u$.

If the norm is given by an inner product then we have $\|u+v\|^{2}=\|u\|^{2}+2 \langle u, v \rangle+\|u\|^{2}$ in the case of real scalars and $\|u+v\|^{2}=\|u\|^{2}+2 \Re \langle u, v \rangle+\|u\|^{2}$ in the case of complex scalars.

Answer 2

It's not true, just take $u=-v$.

Answer 3

$$\|u+v\|^2=(u+v)^2=u^2+2uv+v^2=\|u\|^2+2uv+\|v\|$$

but

$$uv\ne\|u\|\|v\|.$$