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: Understanding the Concept:
Two vectors are perpendicular when their dot product is zero.

Step 2: Set up.
\[ (\vec{A} + \vec{B})\cdot(\vec{A} - \vec{B}) = 0 \]

Step 3: Expand.
\[ \vec{A}\cdot\vec{A} - \vec{A}\cdot\vec{B} + \vec{B}\cdot\vec{A} - \vec{B}\cdot\vec{B} = A^2 - B^2 = 0 \]
The cross terms cancel because the dot product is commutative. So \(A = B\).

Step 4: Check the options.
Only (D), equal in magnitude, follows. The vectors need not be perpendicular or parallel, and they cannot be unequal.

Final Answer:
The two forces are equal in magnitude, option (D). \[ \boxed{|\vec{A}| = |\vec{B}|} \]