Question

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

• A. Are equal to each other in magnitude
• B. Are not equal to each other in magnitude
• C. Cannot be predicted
• D. Are equal to each other

Answer 1

Answer: (A)

Step 1: Given that:
Vector sum of two forces is perpendicular to the vector difference of the two forces.

Step 2: Determination of the relation between the two forces:

Let $\overrightarrow{F_1}$ and $\overrightarrow{F_2}$ be the two forces, then

Vector sum is given as $\overrightarrow{F_1}+\overrightarrow{F_2}$

And the vector difference is given as $\overrightarrow{F_1}-\overrightarrow{F_2}$ .

The two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular to each other if the dot product of the vectors is equal to zero. that is;

$\overrightarrow{a}.\overrightarrow{b}=0$

Hence,

$(\overrightarrow{F_1}+\overrightarrow{F_2}).(\overrightarrow{F_1}-\overrightarrow{F_2})=0$

${\left|\overrightarrow{\overrightarrow{F_1}}\right|}^2-{\left|\overrightarrow{\overrightarrow{F_2}}\right|}^2=0$

${\left|\overrightarrow{\overrightarrow{F_1}}\right|}^2={\left|\overrightarrow{\overrightarrow{F_2}}\right|}^2$

$\left|\overrightarrow{\overrightarrow{F_1}}\right|=\left|\overrightarrow{\overrightarrow{F_2}}\right|$

Thus,

It can be seen that the magnitude of both the forces are equal in this case.

Hence,

Option A) forces are equal to each other in magnitude is the correct option.