Simplify ||x+y||^2 -||x||^2-||y||^2 using dot products
Your solutions are wrong since you didn't transpose.
The standard inner product for $\displaystyle \mathbb{R}$ is the scalar product
$\displaystyle <x,y>=x^T\cdot y$.
$\displaystyle x_{n,1}\cdot x_{n,1}=DNE \ \ \mbox{but} \ x_{1,n}\cdot x_{n,1}=\alpha \ \ \alpha\in\mathbb{R}$
You are only correct if the vectors are 1,1 but you can't make that assumption.
Dot product is not a matrix product of a row and column vector (so no transpose required). Plato's approach is correct and it is the approach that is expected (Your inner product is not a "dot" product as required by the question. You are assuming more knowledge on the part of the OP and so what you post is probably gobbledygook to them)
CB