I have to prove that
\begin{equation*} ||A-B||=||A+B||\Leftrightarrow AB=0 \end{equation*}
and I was wondering if this approach is correct, or if there's a better/more elegant way to prove this.
Given n-dimensional vectors A and B, we can write $||A-B||=||A+B||$ as:
\begin{equation*} \sqrt{\sum\limits_{j=1}^{n}(a_j-b_j)^2}=\sqrt{\sum\limits_{j=1}^{n}(a_j+b_j)^2}. \end{equation*}
Squaring both sides and expanding the binomials:
\begin{equation*} \sum\limits_{j=1}^{n}a^2_j-2a_jb_j+b_j^2=\sum\limits_{j=1}^{n}a^2_j+2a_jb_j+b_j^2. \end{equation*}
Simplifying:
\begin{equation*} -\sum\limits_{j=1}^{n}a_jb_j=\sum\limits_{j=1}^{n}a_jb_j,~\text{which holds true if and only if}~\sum\limits_{j=1}^{n}a_jb_j=0. \end{equation*}
Since $AB$ is equivalent to $\sum\limits_{j=1}^{n}a_jb_j$, then $||A-B||=||A+B||\Leftrightarrow AB=0$
Thanks in advance.