Physics - Vectors Question with Solution | TestHub

Given that the vector sum of the two forces is perpendicular to their vector difference.

Therefore, the dot product of their sum and difference must be zero:

$(\overrightarrow{A}+\overrightarrow{B})\cdot (\overrightarrow{A}-\overrightarrow{B})=0$

Expanding the dot product, we get:

$\overrightarrow{A}\cdot \overrightarrow{A}-\overrightarrow{A}\cdot \overrightarrow{B}+\overrightarrow{B}\cdot \overrightarrow{A}-\overrightarrow{B}\cdot \overrightarrow{B}=0$

Since the dot product is commutative ($\overrightarrow{A}\cdot \overrightarrow{B}=\overrightarrow{B}\cdot \overrightarrow{A}$), the middle terms cancel out:

$|\overrightarrow{A}{|}^2-|\overrightarrow{B}{|}^2=0$

$|\overrightarrow{A}{|}^2=|\overrightarrow{B}{|}^2$

$|\overrightarrow{A}|=|\overrightarrow{B}|$

Hence, the forces $\overrightarrow{A}$ and $\overrightarrow{B}$ are equal in magnitude.

Answer: equal in magnitude.