The vector sum of two forces is perpendicular to their vector differences. In that case, the forces

Solution :
Let $$A$$ and $$B$$ be two forces. The sum of the two forces.
$${F_1} = A + B\,......\left( {\text{i}} \right)$$
The difference of the two forces,
$${F_2} = A - B\,......\left( {{\text{ii}}} \right)$$
Since, sum of the two forces is perpendicular to their differences as given, so
$$\eqalign{ & {F_1} \cdot {F_2} = 0 \cr & {\text{or}}\,\left( {A + B} \right) \cdot \left( {A - B} \right) = 0 \cr & {\text{or}}\,{A^2} - A \cdot B + B \cdot A - {B^2} = 0 \cr & {\text{or}}\,{A^2} = {B^2} \cr & {\text{or}}\,\left| A \right| = \left| B \right| \cr} $$
Thus, the forces are equal to each other in magnitude.