If the resultant of three forces `vec(F)_(1)=phat(i)+3hat(j)-hat(k),vec(F)_(2)=-5hat(i)+hat(j)+2hat(k)` and `vec(F)_(3)=6hat(i)-hat(k)` acting on a particle has magnitude equal to 5 units, then the value (s) of p is (are)

Options

A. `-6`
B. `-4`
C. 2
D. 4

Explanation

To solve the problem, we need to find the value(s) 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. **Write down the forces:**
\[ \vec{F}_1 = p \hat{i} + 3 \hat{j} - \hat{k} \]
\[ \vec{F}_2 = -5 \hat{i} + \hat{j} + 2 \hat{k} \]
\[ \vec{F}_3 = 6 \hat{i} - \hat{k} \]

2. **Calculate the resultant force \( \vec{F}_{\text{resultant}} \):**
\[ \vec{F}_{\text{resultant}} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 \]
Combine the i, j, and k components:
\[ \text{For } \hat{i} \text{ component: } p - 5 + 6 = p + 1 \]
\[ \text{For } \hat{j} \text{ component: } 3 + 1 = 4 \]
\[ \text{For } \hat{k} \text{ component: } -1 + 2 - 1 = 0 \]
Thus, the resultant force is:
\[ \vec{F}_{\text{resultant}} = (p + 1) \hat{i} + 4 \hat{j} + 0 \hat{k} \]

3. **Find the magnitude of the resultant force:**
The magnitude of \( \vec{F}_{\text{resultant}} \) is given by:
\[ |\vec{F}_{\text{resultant}}| = \sqrt{(p + 1)^2 + 4^2 + 0^2} \]
Simplifying this gives:
\[ |\vec{F}_{\text{resultant}}| = \sqrt{(p + 1)^2 + 16} \]

4. **Set the magnitude equal to 5 units:**
According to the problem, the magnitude is equal to 5:
\[ \sqrt{(p + 1)^2 + 16} = 5 \]

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

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

7. **Take the square root of both sides:**
\[ p + 1 = 3 \quad \text{or} \quad p + 1 = -3 \]

8. **Solve for \( p \):**
- From \( p + 1 = 3 \):
\[ p = 3 - 1 = 2 \]
- From \( p + 1 = -3 \):
\[ p = -3 - 1 = -4 \]

### Final Answer:
The values of \( p \) are \( p = 2 \) and \( p = -4 \).