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

BHUBHU 2007

Solution:

The two vectors must be perpendicular if their dot product must be zero.
Let $\vec{A}$ and $\vec{B}$ be two forces. The sum of the two forces,
$\vec{F}_{1}=\vec{A}+\vec{B}\,\,\,$...(i)
The difference of the two forces,
$\vec{F}_{2}=\vec{A}-\vec{B}$...(ii)
Since, sum of the two forces is perpendicular to their difference as said, so
$\vec{F}_{1} \cdot \vec{F}_{2}=0$
Or $(\vec{A}+\vec{B}) \cdot(\vec{A}-\vec{B})=0$
Or $A^{2}-\vec{A} \cdot \vec{B}+\vec{B} \cdot \vec{A}-B^{2}=0$
Or $A^{2}=B^{2}$
Or $|\vec{A}|=|\vec{B}|$
Thus, the forces are equal to each other in magnitude.