Find the moment of force `vecF = hati + hatj + hatk` acting at point `(-2, 3, 4)` about the point `(1, 2, 3)`.

Explanation

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

### Step 1: Identify the Force Vector
The force vector is given as:
\[ \vec{F} = \hat{i} + \hat{j} + \hat{k} \]

### Step 2: Determine the Position Vectors
We need to find the position vector \(\vec{r}\) from the point about which we are calculating the moment (point \(A(1, 2, 3)\)) to the point where the force is applied (point \(B(-2, 3, 4)\)).

The position vector \(\vec{r}\) can be calculated as:
\[ \vec{r} = \vec{B} - \vec{A} = (-2 - 1) \hat{i} + (3 - 2) \hat{j} + (4 - 3) \hat{k} \]
This simplifies to:
\[ \vec{r} = -3\hat{i} + 1\hat{j} + 1\hat{k} \]

### Step 3: Calculate the Moment of Force
The moment of force \(\vec{M}\) is given by the cross product of the position vector \(\vec{r}\) and the force vector \(\vec{F}\):
\[ \vec{M} = \vec{r} \times \vec{F} \]

Using the determinant method to calculate the cross product:
\[ \vec{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix} \]

### Step 4: Calculate the Determinant
Calculating the determinant:
\[ \vec{M} = \hat{i} \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} -3 & 1 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} -3 & 1 \\ 1 & 1 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = 1 - 1 = 0\)
2. \(\begin{vmatrix} -3 & 1 \\ 1 & 1 \end{vmatrix} = (-3)(1) - (1)(1) = -3 - 1 = -4\)
3. \(\begin{vmatrix} -3 & 1 \\ 1 & 1 \end{vmatrix} = -4\) (same as above)

Substituting these back into the equation for \(\vec{M}\):
\[ \vec{M} = 0\hat{i} - (-4)\hat{j} + (-4)\hat{k} \]
This simplifies to:
\[ \vec{M} = 4\hat{j} - 4\hat{k} \]

### Step 5: Factor out the common term
We can factor out the common term:
\[ \vec{M} = 4(\hat{j} - \hat{k}) \]

### Final Answer
Thus, the moment of force about the point \((1, 2, 3)\) is:
\[ \vec{M} = 4(\hat{j} - \hat{k}) \]