Deduce a result about parallelograms I have a problem with an exercise from David Poole's Linear Algebra: A Modern Introduction.
(a) Prove that $||\mathbf {u} + \mathbf {v}||^2 + ||\mathbf {u} - \mathbf {v}||^2 = 2||\mathbf {u}||^2 + 2||\mathbf {v}||^2$ for all vectors $\mathbf {u}$ and $\mathbf {v}$ in $\mathbb {R}^n$.
(b) Draw a diagram showing $\mathbf {u}$, $\mathbf {v}$, $\mathbf {u} + \mathbf {v}$, $\mathbf {u} - \mathbf {v}$ in $\mathbb {R}^2$ and use (a) to deduce a result about parallelograms.
I did (a) and drew some diagrams, but I can't connect the dots and I don't know what I'm supposed to deduce.
 A: Think of the parallelogram having four vertices $0, u, v$, and $u+v$. The lengths of the two diagonals of the parallelogram are $\| u + v \|$ and $\| u - v \|$. The perimeter of the parallelogram is $2 \| u \| + 2 \| v \|$.
You've shown that the sum of squares of the two diagonals is equal to the sum of the squares of the sides making the perimeter. This is called the parallelogram law.
A: I disagree with the answer by davidlowryduda. He is not taking into account the exponent 2 that appears in the equation. (Edit: His answer has now been corrected.)
Consider the parallelogram he describes. I agree with him that the diagonals have lengths $||u + v||$ and $||u - v||$, and that the side lengths are $||u||$, $||v||$, $||u||$, $||v||$. You should be able to formulate the result as a statement about the lengths of the sides and diagonals of a parallelogram.
However, logically, you should not start with $u$ and $v$ and draw the parallelogram. Instead, you should consider an arbitrary parallelogram $ABCD$, give the names $u$ and $v$ to the vectors $AB$ and $AD$, and then invoke the identity you proved. That way the argument will be applicable to any parallelogram.
