There are two force vectors, one of `5 N` and other of `12 N` at what angle the two vectors be added to get resultant vector of `17 N, 7 N` and `13 N` respectively.

Options

A. `0^(@), 180^(@)` and `90^(@)`
B. `0^(@), 90^(@)` and `180^(@)`
C. `0^(@), 90^(@)` and `90^(@)`
D. `180^(@), 0^(@)` and `90^(@)`

Explanation

To solve the problem of finding the angle between two force vectors of magnitudes \( F_1 = 5 \, \text{N} \) and \( F_2 = 12 \, \text{N} \) that results in different resultant vectors of \( R = 17 \, \text{N} \), \( R = 7 \, \text{N} \), and \( R = 13 \, \text{N} \), we can use the formula for the resultant of two vectors:

\[ R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta \]

### Step-by-Step Solution:

1. **Identify the Given Values**:
- \( F_1 = 5 \, \text{N} \)
- \( F_2 = 12 \, \text{N} \)

2. **Calculate \( F_1^2 \) and \( F_2^2 \)**:
- \( F_1^2 = 5^2 = 25 \)
- \( F_2^2 = 12^2 = 144 \)

3. **Set Up the Resultant Equation**:
- The equation becomes:
\[ R^2 = 25 + 144 + 2 \cdot 5 \cdot 12 \cos \theta \]
- Simplifying gives:
\[ R^2 = 169 + 120 \cos \theta \]

4. **Case 1: Resultant \( R = 17 \, \text{N} \)**:
- Substitute \( R = 17 \):
\[ 17^2 = 169 + 120 \cos \theta \]
- This simplifies to:
\[ 289 = 169 + 120 \cos \theta \]
- Rearranging gives:
\[ 120 \cos \theta = 289 - 169 = 120 \]
- Thus:
\[ \cos \theta = \frac{120}{120} = 1 \]
- Therefore, \( \theta = \cos^{-1}(1) = 0^\circ \).

5. **Case 2: Resultant \( R = 7 \, \text{N} \)**:
- Substitute \( R = 7 \):
\[ 7^2 = 169 + 120 \cos \theta \]
- This simplifies to:
\[ 49 = 169 + 120 \cos \theta \]
- Rearranging gives:
\[ 120 \cos \theta = 49 - 169 = -120 \]
- Thus:
\[ \cos \theta = \frac{-120}{120} = -1 \]
- Therefore, \( \theta = \cos^{-1}(-1) = 180^\circ \).

6. **Case 3: Resultant \( R = 13 \, \text{N} \)**:
- Substitute \( R = 13 \):
\[ 13^2 = 169 + 120 \cos \theta \]
- This simplifies to:
\[ 169 = 169 + 120 \cos \theta \]
- Rearranging gives:
\[ 120 \cos \theta = 169 - 169 = 0 \]
- Thus:
\[ \cos \theta = 0 \]
- Therefore, \( \theta = \cos^{-1}(0) = 90^\circ \).

### Summary of Angles:
- For \( R = 17 \, \text{N} \), \( \theta = 0^\circ \)
- For \( R = 7 \, \text{N} \), \( \theta = 180^\circ \)
- For \( R = 13 \, \text{N} \), \( \theta = 90^\circ \)

### Final Answer:
The angles between the vectors for the respective resultant forces are:
- \( 0^\circ \), \( 180^\circ \), and \( 90^\circ \).