The ratio of maximum and minimum magnitudes of the resultant of two vectors `vec(a)` and `vec(b)` is 3:1. Now, `|vec(a)|` is equal to

Options

A. `|vec(b)|`
B. `2|vec(b)|`
C. `3|vec(b)|`
D. `4|vec(b)|`

Explanation

To solve the problem, we need to find the magnitude of vector \( \vec{a} \) given that the ratio of the maximum and minimum magnitudes of the resultant of two vectors \( \vec{a} \) and \( \vec{b} \) is 3:1.

### Step-by-Step Solution:

1. **Understanding the Resultant Magnitudes**:
- The maximum magnitude of the resultant \( R_{\text{max}} \) of two vectors \( \vec{a} \) and \( \vec{b} \) occurs when the vectors are in the same direction:
\[ R_{\text{max}} = |\vec{a}| + |\vec{b}| \]
- The minimum magnitude of the resultant \( R_{\text{min}} \) occurs when the vectors are in opposite directions:
\[ R_{\text{min}} = ||\vec{a}| - |\vec{b}|| = ||a| - |b|| \]

2. **Setting Up the Ratio**:
- According to the problem, the ratio of maximum to minimum magnitudes is given as:
\[ \frac{R_{\text{max}}}{R_{\text{min}}} = \frac{3}{1} \]
- Substituting the expressions for \( R_{\text{max}} \) and \( R_{\text{min}} \):
\[ \frac{|\vec{a}| + |\vec{b}|}{||\vec{a}| - |\vec{b}||} = 3 \]

3. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ |\vec{a}| + |\vec{b}| = 3 \cdot ||\vec{a}| - |\vec{b}|| \]

4. **Considering Cases**:
- We can consider two cases based on the magnitudes of \( |\vec{a}| \) and \( |\vec{b}| \):
- Case 1: \( |\vec{a}| \geq |\vec{b}| \)
\[ |\vec{a}| + |\vec{b}| = 3(|\vec{a}| - |\vec{b}|) \]
- Case 2: \( |\vec{b}| > |\vec{a}| \) (not applicable here since it leads to contradictions).

5. **Solving Case 1**:
- Rearranging the equation:
\[ |\vec{a}| + |\vec{b}| = 3|\vec{a}| - 3|\vec{b}| \]
- Bringing all terms involving \( |\vec{a}| \) and \( |\vec{b}| \) to one side:
\[ 4|\vec{b}| = 2|\vec{a}| \]
- Simplifying gives:
\[ |\vec{a}| = 2|\vec{b}| \]

6. **Conclusion**:
- The magnitude of vector \( \vec{a} \) is twice the magnitude of vector \( \vec{b} \).

### Final Answer:
\[ |\vec{a}| = 2 |\vec{b}| \]