If the resultant of three forces `vecF_1 = p hati + 3hatj -hatk, vecF_2 = 6hati-hatk and vecF_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 find the value of \( p \) such that the resultant of the three forces \( \vec{F_1} = p \hat{i} + 3 \hat{j} - \hat{k} \), \( \vec{F_2} = 6 \hat{i} - \hat{k} \), and \( \vec{F_3} = -5 \hat{i} + \hat{j} + 2 \hat{k} \) has a magnitude of 5 units, we can follow these steps:

### Step 1: Find the resultant vector
The resultant vector \( \vec{R} \) is given by the sum of the three forces:
\[ \vec{R} = \vec{F_1} + \vec{F_2} + \vec{F_3} \]
Substituting the values of the forces:
\[ \vec{R} = (p \hat{i} + 3 \hat{j} - \hat{k}) + (6 \hat{i} - \hat{k}) + (-5 \hat{i} + \hat{j} + 2 \hat{k}) \]

### Step 2: Combine like terms
Now, we combine the \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) components:
- For \( \hat{i} \): \( p + 6 - 5 = p + 1 \)
- For \( \hat{j} \): \( 3 + 1 = 4 \)
- For \( \hat{k} \): \( -1 - 1 + 2 = 0 \)

Thus, the resultant vector is:
\[ \vec{R} = (p + 1) \hat{i} + 4 \hat{j} + 0 \hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
The magnitude of the resultant vector \( \vec{R} \) is given by:
\[ |\vec{R}| = \sqrt{(p + 1)^2 + 4^2} \]
This simplifies to:
\[ |\vec{R}| = \sqrt{(p + 1)^2 + 16} \]

### Step 4: Set the magnitude equal to 5
We know that the magnitude of the resultant vector is equal to 5 units:
\[ \sqrt{(p + 1)^2 + 16} = 5 \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (p + 1)^2 + 16 = 25 \]

### Step 6: Solve for \( (p + 1)^2 \)
Subtract 16 from both sides:
\[ (p + 1)^2 = 9 \]

### Step 7: Take the square root of both sides
Taking the square root gives:
\[ p + 1 = \pm 3 \]

### Step 8: Solve for \( p \)
This results in two equations:
1. \( p + 1 = 3 \) which gives \( p = 2 \)
2. \( p + 1 = -3 \) which gives \( p = -4 \)

Thus, the possible values of \( p \) are:
\[ p = 2 \quad \text{or} \quad p = -4 \]

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

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