What is the magnitude of the moment of the couple consisting of the force `vec(F)=3 hat(i)+2hat(j)-hat(k)` acting through the point `hat(i)-hat(j)+hat(k) and -vec(F)` acting through the point `2hat(i)-3hat(j)-hat(k)`?

Options

A. `2sqrt(5)`
B. `3sqrt(5)`
C. `5sqrt(5)`
D. `7sqrt(5)`

Explanation

To find the magnitude of the moment of the couple consisting of the force \(\vec{F} = 3\hat{i} + 2\hat{j} - \hat{k}\) acting through the point \(\hat{i} - \hat{j} + \hat{k}\) and \(-\vec{F}\) acting through the point \(2\hat{i} - 3\hat{j} - \hat{k}\), we can follow these steps:

### Step 1: Identify the position vectors
Let:
- \(\vec{r_1} = \hat{i} - \hat{j} + \hat{k}\)
- \(\vec{r_2} = 2\hat{i} - 3\hat{j} - \hat{k}\)

### Step 2: Calculate the vector between the two points
The vector \(\vec{r_{12}}\) from \(\vec{r_1}\) to \(\vec{r_2}\) is given by:
\[ \vec{r_{12}} = \vec{r_2} - \vec{r_1} = (2\hat{i} - 3\hat{j} - \hat{k}) - (\hat{i} - \hat{j} + \hat{k}) \]
Calculating this gives:
\[ \vec{r_{12}} = (2 - 1)\hat{i} + (-3 + 1)\hat{j} + (-1 - 1)\hat{k} = \hat{i} - 2\hat{j} - 2\hat{k} \]

### Step 3: Calculate the moment of the couple
The moment of the couple \(\vec{M}\) is given by:
\[ \vec{M} = \vec{r_{12}} \times \vec{F} \]
Substituting \(\vec{F}\):
\[ \vec{M} = (\hat{i} - 2\hat{j} - 2\hat{k}) \times (3\hat{i} + 2\hat{j} - \hat{k}) \]

### Step 4: Use the determinant method to calculate the cross product
We can set up the determinant as follows:
\[ \vec{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -2 & -2 \\ 3 & 2 & -1 \end{vmatrix} \]

Calculating this determinant:
\[ \vec{M} = \hat{i} \begin{vmatrix} -2 & -2 \\ 2 & -1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & -2 \\ 3 & -1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -2 \\ 3 & 2 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} -2 & -2 \\ 2 & -1 \end{vmatrix} = (-2)(-1) - (-2)(2) = 2 + 4 = 6\)

2. \(\begin{vmatrix} 1 & -2 \\ 3 & -1 \end{vmatrix} = (1)(-1) - (-2)(3) = -1 + 6 = 5\)

3. \(\begin{vmatrix} 1 & -2 \\ 3 & 2 \end{vmatrix} = (1)(2) - (-2)(3) = 2 + 6 = 8\)

Putting it all together:
\[ \vec{M} = 6\hat{i} - 5\hat{j} + 8\hat{k} \]

### Step 5: Calculate the magnitude of the moment vector
The magnitude of \(\vec{M}\) is given by:
\[ |\vec{M}| = \sqrt{6^2 + (-5)^2 + 8^2} = \sqrt{36 + 25 + 64} = \sqrt{125} = 5\sqrt{5} \]

### Final Answer
The magnitude of the moment of the couple is \(5\sqrt{5}\).

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