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

Answer

To solve the problem of finding the moment of a force about a given point, we can follow these steps:

### Step 1: Identify the vectors
We are given:
- The point about which we need to find the moment: \( A = \hat{i} + 2\hat{j} - \hat{k} \)
- The point through which the force acts: \( P = 2\hat{i} + 3\hat{j} + \hat{k} \)
- The force vector: \( F = \hat{i} + 2\hat{j} + \hat{k} \)

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

### Step 3: Calculate the moment \( \vec{M} \)
The moment \( \vec{M} \) about point \( A \) due to the force \( F \) is given by the cross product:
\[ \vec{M} = \vec{r} \times \vec{F} \]
Substituting the vectors:
\[ \vec{M} = (\hat{i} + \hat{j} + 2\hat{k}) \times (\hat{i} + 2\hat{j} + \hat{k}) \]

### Step 4: Set up the determinant for the cross product
The cross product can be calculated using the determinant:
\[ \vec{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 1 & 2 \\ 1 & 2 & 1 \end{vmatrix} \]

### Step 5: Calculate the determinant
Calculating the determinant:
\[ \vec{M} = \hat{i} \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} \]

Calculating the 2x2 determinants:
1. \( \begin{vmatrix} 1 & 2 \\ 2 & 1 \end{vmatrix} = (1)(1) - (2)(2) = 1 - 4 = -3 \)
2. \( \begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} = (1)(1) - (2)(1) = 1 - 2 = -1 \)
3. \( \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} = (1)(2) - (1)(1) = 2 - 1 = 1 \)

Substituting back into the equation for \( \vec{M} \):
\[ \vec{M} = -3\hat{i} + 1\hat{j} + 1\hat{k} \]

### Step 6: Final result
Thus, the moment about the point \( A \) is:
\[ \vec{M} = -3\hat{i} + \hat{j} + \hat{k} \]