Question

Let \( \theta \) be the angle between two vectors \( \vec{A} \) and \( \vec{B} \). If \( \hat{a}_\perp \) is the unit vector perpendicular to \( \vec{A} \), then the direction of \( \vec{B} - \vec{B} \sin \theta \hat{a}_\perp \) is:

• A. along \( \vec{B} \)
• B. perpendicular to \( \vec{B} \)
• C. along \( \vec{A} \)
• D. perpendicular to \( \vec{A} \)

Hint:

When subtracting a vector’s perpendicular component from the original vector, the resultant vector is aligned with the direction of the vector from which the perpendicular component is subtracted.

Answer 1

The Correct Option is A

Step 1: Let \(\vec{A}\) and \(\vec{B}\) be two vectors with an angle \(\theta\) between them.

Step 2: The term \(\vec{B} - \vec{B} \sin \theta \hat{a}_\perp\) can be interpreted as the component of \(\vec{B}\) along the direction of \(\vec{A}\). This is because \(\hat{a}_\perp\) is the unit vector perpendicular to \(\vec{A}\), and \(\vec{B} \sin \theta \hat{a}_\perp\) represents the perpendicular component of \(\vec{B}\) to \(\vec{A}\).

Step 3: The remaining vector \(\vec{B} - \vec{B} \sin \theta \hat{a}_\perp\) is thus aligned with the direction of \(\vec{A}\). Therefore, the correct direction is along \(\vec{A}\).