If the resultant of three forces `F_(1)=phati+3hatj-hatk,F_(2)=6hati-hatk and F_(3)=-5hati+hatj+2hatk` acting on a particle has a magnitude equal to 5 units, then the value of p is

Options

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

Explanation

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

### Step-by-Step Solution:

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

2. **Calculate the Resultant Force:**
The resultant force \( F_R \) is the vector sum of the three forces:
\[ F_R = F_1 + F_2 + F_3 \]
Breaking it down into components:
- The \( \hat{i} \) component:
\[ p + 6 - 5 = p + 1 \]
- The \( \hat{j} \) component:
\[ 3 + 1 = 4 \]
- The \( \hat{k} \) component:
\[ -1 - 1 + 2 = 0 \]

Thus, the resultant force can be expressed as:
\[ F_R = (p + 1)\hat{i} + 4\hat{j} + 0\hat{k} \]

3. **Find the Magnitude of the Resultant Force:**
The magnitude of the resultant force \( |F_R| \) is given by:
\[ |F_R| = \sqrt{(p + 1)^2 + 4^2 + 0^2} \]
Simplifying this:
\[ |F_R| = \sqrt{(p + 1)^2 + 16} \]

4. **Set the Magnitude Equal to 5:**
Since we know the magnitude of the resultant force is 5 units, we set up the equation:
\[ \sqrt{(p + 1)^2 + 16} = 5 \]

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

6. **Solve for \( p \):**
Rearranging the equation:
\[ (p + 1)^2 = 25 - 16 \]
\[ (p + 1)^2 = 9 \]
Taking the square root of both sides:
\[ p + 1 = 3 \quad \text{or} \quad p + 1 = -3 \]
Solving these equations:
- For \( p + 1 = 3 \):
\[ p = 2 \]
- For \( p + 1 = -3 \):
\[ p = -4 \]

7. **Final Values of \( p \):**
The possible values of \( p \) are:
\[ p = 2 \quad \text{or} \quad p = -4 \]