Question regarding the algebra of norms

Question

I am new to linear algebra, as am having some doubts regarding the following question:

True or False

$u,v \in R^n$

1. $\left\lVert u\right\rVert=\left\lVert v\right\rVert$ if and only if u+v and u-v are orthogonal.
2. For every $u,v \in R^n$ and every $c \in R$: $\left\lVert cu+v\right\rVert^2=c^2\left\lVert u\right\rVert^2+2c(u·v)+\left\lVert v\right\rVert^2$

I worked out with basic algebra that 1 is true. However I am doubting re 2: algebraically it makes sense, and I have put in two sets of numbers that both came out correct, but I am doubting whether there might be a detail that I am missing (because the question states "for every u,v…"

Thank you!

Answer 1

Just using the definition of the norm in $\mathbb{R}^n$: for $u \in \mathbb{R}^n$ $||u||:=\sqrt{(u,u)}$, where $(\cdot,\cdot)$ represents the standard inner product in $\mathbb{R}^n$.
Now $||cu+v||^2={\sqrt{(cu+v,cu+v)}}^2=(cu+v,cu+v)=(cu+v,cu)+(cu+v,v)=(cu,cu)+(v,cu)+(cu,v)+(v,v)=||cu||^2+(cu,v)+(cu,v)+||v||^2=c^2||u||^2+2c(u,v)+||v||^2$
using the bilinearity and scalar multiplication of the inner product.