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)|`

Answer

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 of Two Vectors**:
The resultant \( R \) of two vectors \( \vec{a} \) and \( \vec{b} \) can be expressed using the formula:
\[ R = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2 |\vec{a}| |\vec{b}| \cos \theta} \]
where \( \theta \) is the angle between the two vectors.

2. **Finding Maximum Resultant**:
The maximum resultant occurs when \( \cos \theta = 1 \) (i.e., when the vectors are in the same direction, \( \theta = 0^\circ \)):
\[ R_{\text{max}} = |\vec{a}| + |\vec{b}| \]

3. **Finding Minimum Resultant**:
The minimum resultant occurs when \( \cos \theta = -1 \) (i.e., when the vectors are in opposite directions, \( \theta = 180^\circ \)):
\[ R_{\text{min}} = ||\vec{a}| - |\vec{b}|| = ||a| - |b|| \]

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

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

6. **Considering Cases**:
We can consider two cases based on the magnitudes of \( |\vec{a}| \) and \( |\vec{b}| \):
- Case 1: \( |\vec{a}| \geq |\vec{b}| \)
\[ |\vec{a}| + |\vec{b}| = 3 (|\vec{a}| - |\vec{b}|) \]
Simplifying this:
\[ |\vec{a}| + |\vec{b}| = 3|\vec{a}| - 3|\vec{b}| \]
\[ 4|\vec{b}| = 2|\vec{a}| \]
\[ |\vec{a}| = 2|\vec{b}| \]

- Case 2: \( |\vec{a}| < |\vec{b}| \)
\[ |\vec{a}| + |\vec{b}| = 3 (|\vec{b}| - |\vec{a}|) \]
This case will lead to a contradiction since it will imply \( |\vec{a}| \) cannot be less than \( |\vec{b}| \) based on the ratio given.

7. **Conclusion**:
Thus, from Case 1, we conclude that:
\[ |\vec{a}| = 2|\vec{b}| \]

### Final Answer:
The magnitude of vector \( \vec{a} \) is \( 2|\vec{b}| \).