Force `hati + 2hatj -3hatk , 2hati + 3hatj + 4hatk ` and `-hati - hatj + hatk` are acting at the point P(0,1,2) . The moment of these forces about the point A(1,-2,0) is

Options

A. `2hati -6hatj + 10hatk`
B. `-2hati + 6hatj - 10 hatk`
C. `2hati + 6hatj -10 hatk`
D. none of these

Answer

To find the moment of the forces acting at point P(0,1,2) about point A(1,-2,0), we can follow these steps:

### Step 1: Identify the Forces
The forces acting at point P are:
- \( \mathbf{F_1} = \hat{i} + 2\hat{j} - 3\hat{k} \)
- \( \mathbf{F_2} = 2\hat{i} + 3\hat{j} + 4\hat{k} \)
- \( \mathbf{F_3} = -\hat{i} - \hat{j} + \hat{k} \)

### Step 2: Calculate the Resultant Force
To find the resultant force \( \mathbf{F_{resultant}} \), we sum the three forces:
\[ \mathbf{F_{resultant}} = \mathbf{F_1} + \mathbf{F_2} + \mathbf{F_3} \]
Calculating the components:
- For \( \hat{i} \): \( 1 + 2 - 1 = 2 \)
- For \( \hat{j} \): \( 2 + 3 - 1 = 4 \)
- For \( \hat{k} \): \( -3 + 4 + 1 = 2 \)

Thus,
\[ \mathbf{F_{resultant}} = 2\hat{i} + 4\hat{j} + 2\hat{k} \]

### Step 3: Determine the Position Vector \( \mathbf{R} \)
The position vector \( \mathbf{R} \) from point A to point P is given by:
\[ \mathbf{R} = \mathbf{P} - \mathbf{A} = (0, 1, 2) - (1, -2, 0) \]
Calculating the components:
- For \( \hat{i} \): \( 0 - 1 = -1 \)
- For \( \hat{j} \): \( 1 - (-2) = 3 \)
- For \( \hat{k} \): \( 2 - 0 = 2 \)

Thus,
\[ \mathbf{R} = -\hat{i} + 3\hat{j} + 2\hat{k} \]

### Step 4: Calculate the Moment \( \mathbf{M} \)
The moment \( \mathbf{M} \) about point A due to the resultant force is given by the cross product:
\[ \mathbf{M} = \mathbf{R} \times \mathbf{F_{resultant}} \]
Using the determinant method to calculate the cross product:
\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -1 & 3 & 2 \\ 2 & 4 & 2 \end{vmatrix} \]

Calculating the determinant:
\[ \mathbf{M} = \hat{i}(3 \cdot 2 - 2 \cdot 4) - \hat{j}(-1 \cdot 2 - 2 \cdot 2) + \hat{k}(-1 \cdot 4 - 3 \cdot 2) \]
Calculating each component:
- For \( \hat{i} \): \( 6 - 8 = -2 \)
- For \( \hat{j} \): \( -(-2 - 4) = 6 \)
- For \( \hat{k} \): \( -(-4 - 6) = -10 \)

Thus,
\[ \mathbf{M} = -2\hat{i} + 6\hat{j} - 10\hat{k} \]

### Final Answer
The moment of these forces about point A is:
\[ \mathbf{M} = -2\hat{i} + 6\hat{j} - 10\hat{k} \]

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