What is the moment about the point `hat(i) + 2hat(j) - hat(k)` of a force represented by `3 hat(i) + hat(k)` acting through the point `2 hat(i) - hat(j) + 3hat(k)`?

Options

A. `-3hat(i) + 11hat(j) + 9 hat(k)`
B. `3hat(i)+ 2hat(j) +9hat(k)`
C. `3hat(i) + 4hat(j) + 9hat(k)`
D. `hat(i) + hat(j) + hat(k)`

Answer

To find the moment (or torque) about the point \( \hat{i} + 2\hat{j} - \hat{k} \) of a force represented by \( 3\hat{i} + \hat{k} \) acting through the point \( 2\hat{i} - \hat{j} + 3\hat{k} \), we can follow these steps:

### Step 1: Define the vectors
Let:
- Point A (where the moment is calculated) be \( \mathbf{A} = \hat{i} + 2\hat{j} - \hat{k} \)
- Point B (where the force is applied) be \( \mathbf{B} = 2\hat{i} - \hat{j} + 3\hat{k} \)
- Force vector \( \mathbf{F} = 3\hat{i} + \hat{k} \)

### Step 2: Calculate the position vector \( \mathbf{r} \)
The position vector \( \mathbf{r} \) from point A to point B is given by:
\[ \mathbf{r} = \mathbf{B} - \mathbf{A} \]
Calculating this:
\[ \mathbf{r} = (2\hat{i} - \hat{j} + 3\hat{k}) - (\hat{i} + 2\hat{j} - \hat{k}) \]
\[ = (2 - 1)\hat{i} + (-1 - 2)\hat{j} + (3 + 1)\hat{k} \]
\[ = \hat{i} - 3\hat{j} + 4\hat{k} \]

### Step 3: Calculate the moment (torque) \( \mathbf{M} \)
The moment (or torque) \( \mathbf{M} \) is given by the cross product:
\[ \mathbf{M} = \mathbf{r} \times \mathbf{F} \]
Substituting \( \mathbf{r} \) and \( \mathbf{F} \):
\[ \mathbf{M} = (\hat{i} - 3\hat{j} + 4\hat{k}) \times (3\hat{i} + \hat{k}) \]

### Step 4: Set up the determinant for the cross product
Using the determinant method for the cross product:
\[ \mathbf{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -3 & 4 \\ 3 & 0 & 1 \end{vmatrix} \]

### Step 5: Calculate the determinant
Calculating the determinant:
\[ \mathbf{M} = \hat{i} \begin{vmatrix} -3 & 4 \\ 0 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 4 \\ 3 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & -3 \\ 3 & 0 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} -3 & 4 \\ 0 & 1 \end{vmatrix} = (-3)(1) - (0)(4) = -3 \)
2. \( \begin{vmatrix} 1 & 4 \\ 3 & 1 \end{vmatrix} = (1)(1) - (3)(4) = 1 - 12 = -11 \)
3. \( \begin{vmatrix} 1 & -3 \\ 3 & 0 \end{vmatrix} = (1)(0) - (3)(-3) = 0 + 9 = 9 \)

Putting it all together:
\[ \mathbf{M} = -3\hat{i} + 11\hat{j} + 9\hat{k} \]

### Final Answer
The moment about the point \( \hat{i} + 2\hat{j} - \hat{k} \) is:
\[ \mathbf{M} = -3\hat{i} + 11\hat{j} + 9\hat{k} \]