Find the moment about `(1,-1,-1)` of the force `3hat(i)+4hat(j)-5hat(k)` acting at `(1, 0,-2)`.

Answer

To find the moment of the force \( \mathbf{F} = 3\hat{i} + 4\hat{j} - 5\hat{k} \) acting at the point \( A(1, 0, -2) \) about the point \( B(1, -1, -1) \), we can follow these steps:

### Step 1: Determine the position vectors
The position vector of point \( A \) is:
\[ \mathbf{A} = 1\hat{i} + 0\hat{j} - 2\hat{k} = \hat{i} - 2\hat{k} \]
The position vector of point \( B \) is:
\[ \mathbf{B} = 1\hat{i} - 1\hat{j} - 1\hat{k} = \hat{i} - \hat{j} - \hat{k} \]

### Step 2: Calculate the vector \( \mathbf{R} \)
The vector \( \mathbf{R} \) from point \( B \) to point \( A \) is given by:
\[ \mathbf{R} = \mathbf{A} - \mathbf{B} \]
Calculating this:
\[ \mathbf{R} = (\hat{i} - 2\hat{k}) - (\hat{i} - \hat{j} - \hat{k}) = \hat{i} - 2\hat{k} - \hat{i} + \hat{j} + \hat{k} \]
Simplifying this gives:
\[ \mathbf{R} = 0\hat{i} + 1\hat{j} - 1\hat{k} = \hat{j} - \hat{k} \]

### Step 3: Calculate the moment \( \mathbf{M} \)
The moment \( \mathbf{M} \) about point \( B \) due to the force \( \mathbf{F} \) is given by the cross product:
\[ \mathbf{M} = \mathbf{R} \times \mathbf{F} \]
Substituting \( \mathbf{R} = \hat{j} - \hat{k} \) and \( \mathbf{F} = 3\hat{i} + 4\hat{j} - 5\hat{k} \):
\[ \mathbf{M} = (\hat{j} - \hat{k}) \times (3\hat{i} + 4\hat{j} - 5\hat{k}) \]

### Step 4: Compute the cross product
Using the determinant method to compute the cross product:
\[ \mathbf{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 1 & -1 \\ 3 & 4 & -5 \end{vmatrix} \]
Calculating this determinant:
1. For \( \hat{i} \):
\[ = \hat{i} \left( 1 \cdot (-5) - (-1) \cdot 4 \right) = \hat{i} (-5 + 4) = -1\hat{i} \]
2. For \( \hat{j} \):
\[ = -\hat{j} \left( 0 \cdot (-5) - (-1) \cdot 3 \right) = -\hat{j} (0 + 3) = -3\hat{j} \]
3. For \( \hat{k} \):
\[ = \hat{k} \left( 0 \cdot 4 - 1 \cdot 3 \right) = \hat{k} (0 - 3) = -3\hat{k} \]

Combining these results:
\[ \mathbf{M} = -1\hat{i} - 3\hat{j} - 3\hat{k} \]

### Step 5: Find the magnitude of the moment
The magnitude of the moment \( |\mathbf{M}| \) is given by:
\[ |\mathbf{M}| = \sqrt{(-1)^2 + (-3)^2 + (-3)^2} = \sqrt{1 + 9 + 9} = \sqrt{19} \]

### Final Result
The moment about the point \( (1, -1, -1) \) of the force \( 3\hat{i} + 4\hat{j} - 5\hat{k} \) is:
\[ \mathbf{M} = -1\hat{i} - 3\hat{j} - 3\hat{k} \quad \text{and} \quad |\mathbf{M}| = \sqrt{19} \]