Chapter 12 - Vectors and the Geometry of Space - 12.3 The Dot Product - 12.3 Exercises - Page 854: 64

$(u+v) $ and $(u-v)$ are orthogonal this implies $(u+v) \cdot (u-v)=0$, then the vectors $u$ and $v$ must have the same length.

$(u+v) $ and $(u-v)$ are orthogonal when $(u+v) \cdot (u-v)=0$ $u \cdot u-u.v+v.u-v.v=0$ $u.u-v.v=0$ Since,$u.u=|u|^2$ Thus, $|u|^2=|v|^2$ Hence, it has proved that $(u+v) $ and $(u-v)$ are orthogonal this implies that $(u+v) \cdot (u-v)=0$, then the vectors $u$ and $v$ must have the same length.