Answer
Using the result in Exercise 85:
$||{\bf{v}} + {\bf{w}}|{|^2} - ||{\bf{v}} - {\bf{w}}|{|^2} = 4{\bf{v}}\cdot{\bf{w}}$
we obtain ${\bf{v}}$ and ${\bf{w}}$ are orthogonal if and only if $||{\bf{v}} - {\bf{w}}|| = ||{\bf{v}} + {\bf{w}}||$.
Work Step by Step
From Exercise 85 we have $||{\bf{v}} + {\bf{w}}|{|^2} - ||{\bf{v}} - {\bf{w}}|{|^2} = 4{\bf{v}}\cdot{\bf{w}}$.
If ${\bf{v}}$ and ${\bf{w}}$ are orthogonal, then ${\bf{v}}\cdot{\bf{w}} = 0$. So,
$||{\bf{v}} + {\bf{w}}|{|^2} - ||{\bf{v}} - {\bf{w}}|{|^2} = 4{\bf{v}}\cdot{\bf{w}} = 0$
$||{\bf{v}} + {\bf{w}}|{|^2} = ||{\bf{v}} - {\bf{w}}|{|^2}$
$||{\bf{v}} + {\bf{w}}|| = ||{\bf{v}} - {\bf{w}}||$
Conversely, if $||{\bf{v}} + {\bf{w}}|| = ||{\bf{v}} - {\bf{w}}||$, then
$||{\bf{v}} + {\bf{w}}|{|^2} = ||{\bf{v}} - {\bf{w}}|{|^2}$
$||{\bf{v}} + {\bf{w}}|{|^2} - ||{\bf{v}} - {\bf{w}}|{|^2} = 0$
But from Exercise 85 we have $||{\bf{v}} + {\bf{w}}|{|^2} - ||{\bf{v}} - {\bf{w}}|{|^2} = 4{\bf{v}}\cdot{\bf{w}}$.
Therefore, ${\bf{v}}\cdot{\bf{w}} = 0$. This implies that ${\bf{v}}$ and ${\bf{w}}$ are orthogonal.
Hence, ${\bf{v}}$ and ${\bf{w}}$ are orthogonal if and only if $||{\bf{v}} - {\bf{w}}|| = ||{\bf{v}} + {\bf{w}}||$.
