Question

The vector sum of two forces $\vec{A}$ and $\vec{B}$ is perpendicular to their vector difference. Hence forces $\vec{A}$ and $\vec{B}$ are}

• A. perpendicular to each other.
• B. parallel to each other.
• C. unequal in magnitude.
• D. equal in magnitude.

Hint:

Sum and Difference vectors are perpendicular only for equal magnitude vectors (diagonals of a rhombus).

Answer 1

The Correct Option is D

Step 1: Concept
If two vectors are perpendicular, their dot product is zero.

Step 2: Analysis
$(\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0$.

Step 3: Calculation
$\vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} = 0$
$A^2 - B^2 = 0 \implies A^2 = B^2 \implies |\vec{A}| = |\vec{B}|$.

Step 4: Conclusion
Hence, the forces are equal in magnitude. Final Answer: (D)