Question

The resultant of two vectors $\vec{A}$ and $\vec{B}$ is $\vec{C}$. If the magnitude of $\vec{B}$ is doubled, the new resultant vector becomes perpendicular to $\vec{A}$, then the magnitude of $\vec{C}$ is

• A. 4 B
• B. 3 B
• C. B
• D. 2 B

Hint:

Use the dot product property $A \cdot B = 0$ whenever you see "perpendicular" in vector problems.

Answer 1

The Correct Option is C

Step 1: Concept
The resultant $\vec{C}$ of two vectors is given by $\vec{C} = \vec{A} + \vec{B}$. When a resultant is perpendicular to a vector, their dot product is zero.

Step 2: Meaning
Let the new resultant be $\vec{R'} = \vec{A} + 2\vec{B}$. Given $\vec{R'} \perp \vec{A}$, so $(\vec{A} + 2\vec{B}) \cdot \vec{A} = 0$.

Step 3: Analysis
Expanding the dot product: $A^2 + 2AB \cos \theta = 0$, which gives $\cos \theta = -A/2B$. Now, the magnitude of the original resultant $C$ is $C^2 = A^2 + B^2 + 2AB \cos \theta$. Substituting $2AB \cos \theta = -A^2$, we get $C^2 = A^2 + B^2 - A^2 = B^2$.

Step 4: Conclusion
Therefore, $C = B$. Final Answer: (C)