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

Hint:

When two vectors are perpendicular, the magnitude of their resultant is \( \sqrt{A^2 + B^2} \).

Answer 1

The Correct Option is C

Step 1: Vector addition formula.
The magnitude of the resultant vector \( \vec{C} \) of two vectors \( \vec{A} \) and \( \vec{B} \) is given by the formula: \[ |\vec{C}| = \sqrt{A^2 + B^2 + 2AB \cos \theta} \] where \( \theta \) is the angle between \( \vec{A} \) and \( \vec{B} \).
Step 2: Doubling \( \vec{B} \).
If the magnitude of \( \vec{B} \) is doubled, the new resultant vector \( \vec{C'} \) will be perpendicular to \( \vec{A} \). This means \( \theta = 90^\circ \), so \( \cos 90^\circ = 0 \), and the formula simplifies to: \[ |\vec{C'}| = \sqrt{A^2 + B^2} \] Since \( B \) is doubled, the magnitude of the new resultant vector is \( |\vec{C'}| = B \), which means the magnitude of \( \vec{C} \) is \( B \).

Step 3: Conclusion.
The magnitude of \( \vec{C} \) is \( B \), so the correct answer is (C).