A force `vec(F)=4hat(i)+hat(k)` acts through point A (0, 2, 0). Find the moment `vec(m)` of `vec(F)` about the point B (4, 0, 4).

Answer

To find the moment \(\vec{m}\) of the force \(\vec{F}\) about the point B, we will follow these steps:

### Step 1: Identify the given information
We have:
- Force vector: \(\vec{F} = 4\hat{i} + \hat{k}\)
- Point A: \(A(0, 2, 0)\)
- Point B: \(B(4, 0, 4)\)

### Step 2: Find the position vector \(\vec{r}\)
The position vector \(\vec{r}\) from point A to point B is given by:
\[ \vec{r} = \vec{B} - \vec{A} \]
Calculating the components:
\[ \vec{A} = 0\hat{i} + 2\hat{j} + 0\hat{k} = 0\hat{i} + 2\hat{j} + 0\hat{k} \]
\[ \vec{B} = 4\hat{i} + 0\hat{j} + 4\hat{k} \]
Thus,
\[ \vec{r} = (4\hat{i} + 0\hat{j} + 4\hat{k}) - (0\hat{i} + 2\hat{j} + 0\hat{k}) = 4\hat{i} - 2\hat{j} + 4\hat{k} \]

### Step 3: Calculate the moment \(\vec{m}\)
The moment \(\vec{m}\) of the force about point B is given by the cross product:
\[ \vec{m} = \vec{r} \times \vec{F} \]
Substituting the values of \(\vec{r}\) and \(\vec{F}\):
\[ \vec{m} = (4\hat{i} - 2\hat{j} + 4\hat{k}) \times (4\hat{i} + 0\hat{j} + 1\hat{k}) \]

### Step 4: Set up the determinant for the cross product
Using the determinant method:
\[ \vec{m} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -2 & 4 \\ 4 & 0 & 1 \end{vmatrix} \]

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

Calculating each of the minors:
1. For \(\hat{i}\):
\[ \begin{vmatrix} -2 & 4 \\ 0 & 1 \end{vmatrix} = (-2)(1) - (4)(0) = -2 \]

2. For \(\hat{j}\):
\[ \begin{vmatrix} 4 & 4 \\ 4 & 1 \end{vmatrix} = (4)(1) - (4)(4) = 4 - 16 = -12 \]

3. For \(\hat{k}\):
\[ \begin{vmatrix} 4 & -2 \\ 4 & 0 \end{vmatrix} = (4)(0) - (-2)(4) = 0 + 8 = 8 \]

### Step 6: Combine the results
Putting it all together:
\[ \vec{m} = -2\hat{i} + 12\hat{j} + 8\hat{k} \]

### Final Answer
Thus, the moment \(\vec{m}\) of the force \(\vec{F}\) about point B is:
\[ \vec{m} = -2\hat{i} + 12\hat{j} + 8\hat{k} \]