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

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

AIPMTAIPMT 2003Motion in a Plane

Solution:

Given : $\left(\vec{F}_{1}+\vec{F}_{2}\right) \perp\left(\vec{F}_{1}-\vec{F}_{2}\right)$
$\therefore\left(\vec{F}_{1}+\vec{F}_{2}\right) \cdot\left(\vec{F}_{1}-\vec{F}_{2}\right)=0 $
$F_{1}^{2}-F_{2}^{2}-\vec{F}_{1} \cdot \vec{F}_{2}+\vec{F}_{2} \cdot \vec{F}_{1}=0 $
$\Rightarrow F_{1}^{2}=F_{2}^{2}$
i.e. $F_{1}, F_{2}$ are equal to each other in magnitude.