Two vector having magnitude 8 and 10 can maximum and minimum value of magnitude of their resultant as

Options

A. 12,6
B. 10,3
C. 18,2
D. none of these

Explanation

To solve the problem of finding the maximum and minimum values of the resultant of two vectors with magnitudes 8 and 10, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Magnitudes of the Vectors**:
Let the magnitudes of the two vectors be:
- \( A = 8 \)
- \( B = 10 \)

2. **Formula for Resultant of Two Vectors**:
The magnitude of the resultant vector \( R \) when two vectors are added can be calculated using the formula:
\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]
where \( \theta \) is the angle between the two vectors.

3. **Finding Maximum Resultant**:
To find the maximum resultant, we take \( \cos \theta = 1 \) (which occurs when the vectors are in the same direction):
\[ R_{\text{max}} = \sqrt{A^2 + B^2 + 2AB} \]
Substituting the values:
\[ R_{\text{max}} = \sqrt{8^2 + 10^2 + 2 \cdot 8 \cdot 10 \cdot 1} \]
\[ = \sqrt{64 + 100 + 160} = \sqrt{324} = 18 \]

4. **Finding Minimum Resultant**:
To find the minimum resultant, we take \( \cos \theta = -1 \) (which occurs when the vectors are in opposite directions):
\[ R_{\text{min}} = \sqrt{A^2 + B^2 - 2AB} \]
Substituting the values:
\[ R_{\text{min}} = \sqrt{8^2 + 10^2 - 2 \cdot 8 \cdot 10} \]
\[ = \sqrt{64 + 100 - 160} = \sqrt{4} = 2 \]

5. **Conclusion**:
The maximum and minimum values of the magnitude of the resultant vector are:
- Maximum Resultant: \( 18 \)
- Minimum Resultant: \( 2 \)

### Final Answer:
The maximum and minimum values of the magnitude of their resultant are \( 18 \) and \( 2 \), respectively.

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