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: Understanding the Question:
We use the dot product property for perpendicular vectors.

Step 2: Key Formula or Approach:
If \( \vec{X} \perp \vec{Y} \), then \( \vec{X} \cdot \vec{Y} = 0 \).

Step 3: Detailed Explanation:
Given \( (\vec{A} + \vec{B}) \perp (\vec{A} - \vec{B}) \).
\[ (\vec{A} + \vec{B}) \cdot (\vec{A} - \vec{B}) = 0 \]
\[ \vec{A} \cdot \vec{A} - \vec{A} \cdot \vec{B} + \vec{B} \cdot \vec{A} - \vec{B} \cdot \vec{B} = 0 \]
Since dot product is commutative (\( \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} \)):
\[ |\vec{A}|^2 - |\vec{B}|^2 = 0 \implies A^2 = B^2 \]
\[ A = B \]

Step 4: Final Answer:
The forces are equal in magnitude.