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. \(B\)
• B. \(2B\)
• C. \(3B\)
• D. \(4B\)

Hint:

Use the condition that the new resultant is perpendicular to A, so its dot product with A is zero, and compare the two resultant magnitudes.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
Let the angle between \(\vec{A}\) and \(\vec{B}\) be \(\theta\). The resultant satisfies \(C^{2}=A^{2}+B^{2}+2AB\cos\theta\).

Step 2: Use the perpendicular condition
The new resultant is \(\vec{A}+2\vec{B}\), and it is perpendicular to \(\vec{A}\).
\(\vec{A}\cdot(\vec{A}+2\vec{B})=0\Rightarrow A^{2}+2AB\cos\theta=0\)
So \(2AB\cos\theta=-A^{2}\).

Step 3: Calculation
Substitute into the first relation:
\(C^{2}=A^{2}+B^{2}-A^{2}=B^{2}\)
Hence \(C=B\).

Step 4: Compare the options
The magnitude of \(\vec{C}\) equals \(B\), which is option (A).

Final Answer:
The magnitude of the resultant C equals B. \[ \boxed{\text{(A) }B} \]