For two vectors 
prove that we have the identity
![\[ \lVert A + B\rVert^2 + \lVert A - B \rVert^2 = 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-08fb8b3e378823dcbf4423a6273ab87a_l3.png "Rendered by QuickLaTeX.com")
What geometric theorem can you deduce from this identity?
Proof. We compute
![\begin{align*} \lVert A + B \rVert^2 + \lVert A - B \rVert^2 &= \sum_{i=1}^n (a_i + b_i)^2 + \sum_{i=1}^n (a_i - b_i)^2 \\[9pt] &= \sum_{i=1}^n (2a_i^2 + 2b_i^2) \\[9pt] &= 2 \sum_{i=1}^n a_i^2 + 2 \sum_{i=1}^n b_i^2 \\[9pt] &= 2 \lVert A \rVert^2 + 2 \lVert B \rVert^2. \qquad \blacksquare \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-1473a708330b58d7269c414d8dbeeaaf_l3.png "Rendered by QuickLaTeX.com")
For two vectors prove that we have the identity
What geometric theorem can you deduce from this identity?
Proof. We compute
Is the geometric identity the one that the sum of squares of diagonals equals two times the sum of squares of non paralel sides ? To be more precise this one :- https://en.wikipedia.org/wiki/Parallelogram_law
Yes, as written at the back of the book (i.e. there are 4 sides in total).