The resultant of two vectors makes an angle of ` 60^(@)` with one vector and ` 30^(@)` with one other. If the magnitude of resultant is 25, the two vectors are of magnitude

Options

A. 24,7
B. 12,5
C. 12.5,12.5`sqrt3`
D. `15.7, 15.7 sqrt2`

Answer

To solve the problem, we need to find the magnitudes of two vectors \( \vec{A} \) and \( \vec{B} \) given that their resultant \( \vec{R} \) makes angles of \( 60^\circ \) with \( \vec{A} \) and \( 30^\circ \) with \( \vec{B} \), and the magnitude of the resultant is \( 25 \).

### Step-by-Step Solution:

1. **Understand the Geometry**:
We have two vectors \( \vec{A} \) and \( \vec{B} \) with a resultant \( \vec{R} \). The angles given are:
- \( \angle R \vec{A} = 60^\circ \)
- \( \angle R \vec{B} = 30^\circ \)

2. **Use the Law of Cosines**:
The magnitude of the resultant vector can be expressed in terms of the magnitudes of \( \vec{A} \) and \( \vec{B} \) and the angles between them. However, we can also use the components of the vectors based on the angles provided.

3. **Resolve \( \vec{A} \) and \( \vec{B} \)**:
- The component of \( \vec{A} \) along the direction of \( \vec{R} \) is:
\[ A = R \cdot \cos(60^\circ) = 25 \cdot \frac{1}{2} = 12.5 \]
- The component of \( \vec{B} \) along the direction of \( \vec{R} \) is:
\[ B = R \cdot \sin(60^\circ) = 25 \cdot \frac{\sqrt{3}}{2} = 12.5\sqrt{3} \]

4. **Final Magnitudes**:
- Thus, the magnitudes of the vectors are:
- \( |\vec{A}| = 12.5 \)
- \( |\vec{B}| = 12.5\sqrt{3} \)

5. **Conclusion**:
The magnitudes of the two vectors are \( 12.5 \) and \( 12.5\sqrt{3} \).