Question

The vector sum of the two forces \(\overset{⃗}{A}\) and \(\overset{⃗}{B}\) is perpendicular to their vector difference. Hence forces \(\overset{⃗}{A}\) and \(\overset{⃗}{B}\) are

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

Hint:

Perpendicular vectors have zero dot product, so set the dot product of the sum and difference to zero.

Answer 1

The Correct Option is D

Step 1: Geometric view.
The sum and the difference of two vectors are the diagonals of the parallelogram built on them.

Step 2: Condition.
The diagonals of a parallelogram are perpendicular only when it is a rhombus, which means all sides are equal. So $|\vec{A}| = |\vec{B}|$.

Step 3: Check algebra.
$(\vec{A} + \vec{B})\cdot(\vec{A} - \vec{B}) = A^2 - B^2 = 0$, so $A = B$.

Final Answer:
Option (D). \[ \boxed{|\vec{A}| = |\vec{B}|} \]