A force of `(2 hati - 4 hatj + 2 hatk )`N act a point `(3 hati+2 hatj -4 hatk)` metre form the origin. The magnitude of torque is

Options

A. zero
B. 24.4 N-m
C. 0.244 N-m
D. 2.444 N-m

Answer

To find the magnitude of the torque given a force and a position vector, we can follow these steps:

### Step 1: Identify the vectors
The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 2 \hat{i} - 4 \hat{j} + 2 \hat{k} \, \text{N} \]
The position vector \( \mathbf{r} \) is given as:
\[ \mathbf{r} = 3 \hat{i} + 2 \hat{j} - 4 \hat{k} \, \text{m} \]

### Step 2: Calculate the torque vector
The torque \( \mathbf{\tau} \) is calculated using the cross product of the position vector \( \mathbf{r} \) and the force vector \( \mathbf{F} \):
\[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]

### Step 3: Set up the determinant for the cross product
We can use the determinant of a matrix to calculate the cross product:
\[ \mathbf{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 2 & -4 \\ 2 & -4 & 2 \end{vmatrix} \]

### Step 4: Calculate the determinant
Expanding the determinant, we have:
\[ \mathbf{\tau} = \hat{i} \begin{vmatrix} 2 & -4 \\ -4 & 2 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -4 \\ 2 & 2 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & 2 \\ 2 & -4 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. For \( \hat{i} \):
\[ \begin{vmatrix} 2 & -4 \\ -4 & 2 \end{vmatrix} = (2)(2) - (-4)(-4) = 4 - 16 = -12 \]

2. For \( \hat{j} \):
\[ \begin{vmatrix} 3 & -4 \\ 2 & 2 \end{vmatrix} = (3)(2) - (-4)(2) = 6 + 8 = 14 \]

3. For \( \hat{k} \):
\[ \begin{vmatrix} 3 & 2 \\ 2 & -4 \end{vmatrix} = (3)(-4) - (2)(2) = -12 - 4 = -16 \]

Putting it all together:
\[ \mathbf{\tau} = -12 \hat{i} - 14 \hat{j} - 16 \hat{k} \]

### Step 5: Calculate the magnitude of the torque
The magnitude of the torque \( |\mathbf{\tau}| \) is given by:
\[ |\mathbf{\tau}| = \sqrt{(-12)^2 + (-14)^2 + (-16)^2} \]
Calculating each term:
\[ = \sqrt{144 + 196 + 256} = \sqrt{596} \]

### Step 6: Final calculation
Calculating the square root:
\[ |\mathbf{\tau}| \approx 24.41 \, \text{N m} \]

### Conclusion
The magnitude of the torque is approximately:
\[ \boxed{24.41 \, \text{N m}} \]