A force `F=2hat(i)+hat(j)-hat(k)` acts at point A whose position vector is `2hat(i)-hat(j)`. Find the moment of force F about the origin.

Answer

To find the moment of the force \( \mathbf{F} \) about the origin, we will follow these steps:

### Step 1: Identify the vectors
The force vector \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 2\hat{i} + \hat{j} - \hat{k} \]
The position vector \( \mathbf{A} \) at which the force acts is given as:
\[ \mathbf{A} = 2\hat{i} - \hat{j} \]

### Step 2: Determine the position vector from the origin to point A
The position vector from the origin \( O \) to point \( A \) is simply the vector \( \mathbf{A} \):
\[ \mathbf{R} = \mathbf{A} - \mathbf{O} = (2\hat{i} - \hat{j}) - (0\hat{i} + 0\hat{j} + 0\hat{k}) = 2\hat{i} - \hat{j} \]

### Step 3: Calculate the moment of the force about the origin
The moment \( \mathbf{M} \) of the force about the origin is given by the cross product of the position vector \( \mathbf{R} \) and the force vector \( \mathbf{F} \):
\[ \mathbf{M} = \mathbf{F} \times \mathbf{R} \]
Substituting the vectors:
\[ \mathbf{M} = (2\hat{i} + \hat{j} - \hat{k}) \times (2\hat{i} - \hat{j}) \]

### Step 4: Set up the determinant for the cross product
We can compute the cross product using the determinant of a matrix:
\[ \mathbf{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & -1 \\ 2 & -1 & 0 \end{vmatrix} \]

### Step 5: Calculate the determinant
Expanding the determinant along the first row:
\[ \mathbf{M} = \hat{i} \begin{vmatrix} 1 & -1 \\ -1 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 2 & -1 \\ 2 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
1. For \( \hat{i} \):
\[ \begin{vmatrix} 1 & -1 \\ -1 & 0 \end{vmatrix} = (1)(0) - (-1)(-1) = 0 - 1 = -1 \]
2. For \( \hat{j} \):
\[ \begin{vmatrix} 2 & -1 \\ 2 & 0 \end{vmatrix} = (2)(0) - (-1)(2) = 0 + 2 = 2 \]
3. For \( \hat{k} \):
\[ \begin{vmatrix} 2 & 1 \\ 2 & -1 \end{vmatrix} = (2)(-1) - (1)(2) = -2 - 2 = -4 \]

### Step 6: Combine the results
Putting it all together:
\[ \mathbf{M} = -1\hat{i} - 2\hat{j} - 4\hat{k} \]
Thus, the moment of the force about the origin is:
\[ \mathbf{M} = -\hat{i} - 2\hat{j} - 4\hat{k} \]

### Final Answer
The moment of force \( \mathbf{F} \) about the origin is:
\[ \mathbf{M} = -\hat{i} - 2\hat{j} - 4\hat{k} \]

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