The moment of the force, `vec F = 4 hati + 5 hat j - 6 hat k` at `(2, 0 , - 3)`. About the point `(2, - 2, -2)` is given by

Options

A. `-7hati-8hatj-4hatk`
B. `-4hati-hatj-8hatk`
C. `-8hati-4hatj-7k`
D. `-7hati-4hatj-8hatk`

Answer

To find the moment of the force \( \vec{F} = 4 \hat{i} + 5 \hat{j} - 6 \hat{k} \) at the point \( (2, 0, -3) \) about the point \( (2, -2, -2) \), we can follow these steps:

### Step 1: Define the Points
Let point \( P \) be \( (2, -2, -2) \) and point \( Q \) be \( (2, 0, -3) \).

### Step 2: Calculate the Position Vector \( \vec{R} \)
The position vector \( \vec{R} \) from point \( P \) to point \( Q \) is calculated as:
\[ \vec{R} = \vec{Q} - \vec{P} = (2, 0, -3) - (2, -2, -2) \]
Calculating this gives:
\[ \vec{R} = (2 - 2) \hat{i} + (0 - (-2)) \hat{j} + (-3 - (-2)) \hat{k} = 0 \hat{i} + 2 \hat{j} - 1 \hat{k} \]
Thus,
\[ \vec{R} = 0 \hat{i} + 2 \hat{j} - 1 \hat{k} \]

### Step 3: Write the Force Vector
The force vector is given as:
\[ \vec{F} = 4 \hat{i} + 5 \hat{j} - 6 \hat{k} \]

### Step 4: Calculate the Moment (Torque) \( \tau \)
The moment (or torque) \( \tau \) is given by the cross product:
\[ \tau = \vec{R} \times \vec{F} \]
Using the determinant method, we can set up the cross product as follows:
\[ \tau = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 2 & -1 \\ 4 & 5 & -6 \end{vmatrix} \]

### Step 5: Calculate the Determinant
Calculating the determinant:
\[ \tau = \hat{i} \begin{vmatrix} 2 & -1 \\ 5 & -6 \end{vmatrix} - \hat{j} \begin{vmatrix} 0 & -1 \\ 4 & -6 \end{vmatrix} + \hat{k} \begin{vmatrix} 0 & 2 \\ 4 & 5 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \( \begin{vmatrix} 2 & -1 \\ 5 & -6 \end{vmatrix} = (2)(-6) - (-1)(5) = -12 + 5 = -7 \)
2. \( \begin{vmatrix} 0 & -1 \\ 4 & -6 \end{vmatrix} = (0)(-6) - (-1)(4) = 0 + 4 = 4 \)
3. \( \begin{vmatrix} 0 & 2 \\ 4 & 5 \end{vmatrix} = (0)(5) - (2)(4) = 0 - 8 = -8 \)

Putting it all together:
\[ \tau = -7 \hat{i} - 4 \hat{j} - 8 \hat{k} \]

### Final Result
Thus, the moment of the force about the point \( (2, -2, -2) \) is:
\[ \tau = -7 \hat{i} - 4 \hat{j} - 8 \hat{k} \]