Question

The resultant of two vectors \(\vec{P}\) and \(\vec{Q}\) is \(\vec{R}\). When the direction of \(\vec{Q}\) is reversed, the resultant is given by \(\vec{S}\). Which one of the following is true for vectors \(\vec{R}\) and \(\vec{S}\)?

• A. \( R^2 - S^2 = (P^2 + Q^2) \)
• B. \( R^2 - S^2 = 2(\vec{P}\cdot\vec{Q}) \)
• C. \( R^2 + S^2 = 4(\vec{P}\cdot\vec{Q}) \)
• D. \( R^2 + S^2 = 2(P^2 + Q^2) \)

Hint:

For vectors \(\vec{P}\pm\vec{Q}\), adding squares eliminates the dot product term.

Answer 1

The Correct Option is D

Step 1: Writing the resultants.
\[ \vec{R} = \vec{P} + \vec{Q}, \quad \vec{S} = \vec{P} - \vec{Q}. \]
Step 2: Squaring magnitudes.
\[ R^2 = P^2 + Q^2 + 2\vec{P}\cdot\vec{Q}, \] \[ S^2 = P^2 + Q^2 - 2\vec{P}\cdot\vec{Q}. \]
Step 3: Adding the two equations.
\[ R^2 + S^2 = 2(P^2 + Q^2). \]
Step 4: Conclusion.
Thus, the correct relation is \( R^2 + S^2 = 2(P^2 + Q^2) \).