If the resultant of three forces `vecF_1= phati +3hatj-hatk , vecF_2 =-5hati+hatj+2hatk and vecF_3= 6hati -hatk` acting on a particle has a magnitude equal to 5 units, then what is difference in the values of `p` ?

Explanation

To solve the problem, we need to find the difference in the values of \( p \) such that the resultant of the three forces \( \vec{F_1}, \vec{F_2}, \) and \( \vec{F_3} \) has a magnitude of 5 units.

### Step-by-Step Solution:

1. **Identify the Forces:**
- Given:
\[ \vec{F_1} = p\hat{i} + 3\hat{j} - \hat{k} \]
\[ \vec{F_2} = -5\hat{i} + \hat{j} + 2\hat{k} \]
\[ \vec{F_3} = 6\hat{i} - \hat{k} \]

2. **Calculate the Resultant Force:**
- The resultant force \( \vec{R} \) is the sum of the three forces:
\[ \vec{R} = \vec{F_1} + \vec{F_2} + \vec{F_3} \]
- Combine the \( \hat{i}, \hat{j}, \) and \( \hat{k} \) components:
- For \( \hat{i} \):
\[ p - 5 + 6 = p + 1 \]
- For \( \hat{j} \):
\[ 3 + 1 = 4 \]
- For \( \hat{k} \):
\[ -1 + 2 - 1 = 0 \]
- Thus, the resultant vector is:
\[ \vec{R} = (p + 1)\hat{i} + 4\hat{j} + 0\hat{k} \]

3. **Find the Magnitude of the Resultant Force:**
- The magnitude of \( \vec{R} \) is given by:
\[ |\vec{R}| = \sqrt{(p + 1)^2 + 4^2} \]
- Since the magnitude is given as 5 units:
\[ \sqrt{(p + 1)^2 + 16} = 5 \]

4. **Square Both Sides:**
- Squaring both sides to eliminate the square root:
\[ (p + 1)^2 + 16 = 25 \]

5. **Solve for \( (p + 1)^2 \):**
- Rearranging gives:
\[ (p + 1)^2 = 25 - 16 \]
\[ (p + 1)^2 = 9 \]

6. **Find the Values of \( p + 1 \):**
- Taking the square root of both sides:
\[ p + 1 = 3 \quad \text{or} \quad p + 1 = -3 \]

7. **Calculate the Values of \( p \):**
- From \( p + 1 = 3 \):
\[ p = 2 \]
- From \( p + 1 = -3 \):
\[ p = -4 \]

8. **Find the Difference in Values of \( p \):**
- The difference in the values of \( p \) is:
\[ |2 - (-4)| = |2 + 4| = 6 \]

### Final Answer:
The difference in the values of \( p \) is **6**.