Question

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

• A. are not equal to each other in magnitude
• B. cannot be predicted
• C. are equal to each other
• D. are equal to each other in magnitude

Answer 1

Answer: (D)

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.