Question

The magnitudes of two vectors \(\vec{A}\) and \(\vec{B}\) are \(A\) and \(B\) respectively, the magnitude of their resultant vector \(\vec{R}\) is \(R\). If \(\theta\) is the angle between the vectors \(\vec{A}\) and \(\vec{B}\), the angle made by the vector \(\vec{R}\) with the vector \(\vec{A}\) is \(\alpha\), then

• A. \( R = \frac{B \sin \alpha}{\sin \theta} \)
• B. \( R = \frac{A \sin \alpha}{\sin \theta} \)
• C. \( R = \frac{B \sin \theta}{\sin \alpha} \)
• D. \( R = \frac{A \sin \theta}{\sin \alpha} \)

Hint:

In triangle law of vector addition, use the sine rule: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) where sides and angles are in opposite pairs.

Answer 1

The Correct Option is C

This is a case of vector triangle law. If vectors \( \vec{A} \) and \( \vec{B} \) form the two sides of a triangle and the resultant \( \vec{R} \) is the closing side, the sine rule applies: \[ \frac{R}{\sin \theta} = \frac{B}{\sin \alpha} \Rightarrow R = \frac{B \sin \theta}{\sin \alpha} \] This is derived using triangle geometry in vector addition.