Two vector having magnitude 8 and 10 can maximum and minium 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 vector \( A \) have a magnitude of \( 8 \).
- Let vector \( B \) have a magnitude of \( 10 \).

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

3. **Calculate \( A^2 \) and \( B^2 \)**:
- Calculate \( A^2 = 8^2 = 64 \).
- Calculate \( B^2 = 10^2 = 100 \).

4. **Determine the Maximum and Minimum Values of \( R \)**:
- The maximum value of \( R \) occurs when \( \cos \theta = 1 \) (i.e., the vectors are in the same direction).
\[ R_{\text{max}} = \sqrt{64 + 100 + 2 \cdot 8 \cdot 10 \cdot 1} \]
\[ R_{\text{max}} = \sqrt{64 + 100 + 160} = \sqrt{324} = 18 \]

- The minimum value of \( R \) occurs when \( \cos \theta = -1 \) (i.e., the vectors are in opposite directions).
\[ R_{\text{min}} = \sqrt{64 + 100 - 2 \cdot 8 \cdot 10} \]
\[ R_{\text{min}} = \sqrt{64 + 100 - 160} = \sqrt{4} = 2 \]

5. **Final Result**:
- The maximum value of the magnitude of the resultant is \( 18 \).
- The minimum value of the magnitude of the resultant is \( 2 \).

### Summary:
- Maximum value of the resultant \( R_{\text{max}} = 18 \)
- Minimum value of the resultant \( R_{\text{min}} = 2 \)

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