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^(@)`

Answer

To solve the problem of finding the angle at which two force vectors (5 N and 12 N) should be added to obtain resultant vectors of 17 N, 7 N, and 13 N respectively, we will use the formula for the resultant of two vectors given by:

\[ R = \sqrt{A^2 + B^2 + 2AB \cos \theta} \]

Where:
- \( R \) is the resultant vector
- \( A \) and \( B \) are the magnitudes of the two vectors
- \( \theta \) is the angle between the two vectors

### Step 1: Finding the angle for the resultant of 17 N

1. Given:
- \( A = 5 \, \text{N} \)
- \( B = 12 \, \text{N} \)
- \( R = 17 \, \text{N} \)

2. Substitute the values into the resultant formula:
\[ 17 = \sqrt{5^2 + 12^2 + 2 \cdot 5 \cdot 12 \cos \theta} \]

3. Calculate \( 5^2 + 12^2 \):
\[ 5^2 = 25, \quad 12^2 = 144 \quad \Rightarrow \quad 25 + 144 = 169 \]

4. Now, square both sides:
\[ 17^2 = 169 + 120 \cos \theta \]
\[ 289 = 169 + 120 \cos \theta \]

5. Rearranging gives:
\[ 120 \cos \theta = 289 - 169 \]
\[ 120 \cos \theta = 120 \]

6. Dividing by 120:
\[ \cos \theta = 1 \]

7. Therefore:
\[ \theta = 0^\circ \]

### Step 2: Finding the angle for the resultant of 7 N

1. Given \( R = 7 \, \text{N} \):
\[ 7 = \sqrt{5^2 + 12^2 + 2 \cdot 5 \cdot 12 \cos \theta} \]

2. Substitute the values:
\[ 7^2 = 25 + 144 + 120 \cos \theta \]
\[ 49 = 169 + 120 \cos \theta \]

3. Rearranging gives:
\[ 120 \cos \theta = 49 - 169 \]
\[ 120 \cos \theta = -120 \]

4. Dividing by 120:
\[ \cos \theta = -1 \]

5. Therefore:
\[ \theta = 180^\circ \]

### Step 3: Finding the angle for the resultant of 13 N

1. Given \( R = 13 \, \text{N} \):
\[ 13 = \sqrt{5^2 + 12^2 + 2 \cdot 5 \cdot 12 \cos \theta} \]

2. Substitute the values:
\[ 13^2 = 25 + 144 + 120 \cos \theta \]
\[ 169 = 169 + 120 \cos \theta \]

3. Rearranging gives:
\[ 120 \cos \theta = 169 - 169 \]
\[ 120 \cos \theta = 0 \]

4. Dividing by 120:
\[ \cos \theta = 0 \]

5. Therefore:
\[ \theta = 90^\circ \]

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