Find the cross product `vec(r ) xx vec(F) " given " vec(F ) = hat(i) + hat(j) + hat(k) and vec (r )` is the distance between two points whose coordinates are `(-2,3,4) and (1,2,3)`

Explanation

To find the cross product \(\vec{r} \times \vec{F}\), we first need to determine the vector \(\vec{r}\) based on the coordinates of the two points given.

### Step-by-Step Solution:

1. **Identify the Coordinates of the Points**:
The coordinates of the two points are:
- Point 1: \((-2, 3, 4)\)
- Point 2: \((1, 2, 3)\)

2. **Calculate the Vector \(\vec{r}\)**:
The vector \(\vec{r}\) is defined as the difference between the position vectors of the two points:
\[ \vec{r} = \vec{r_2} - \vec{r_1} \]
Where:
- \(\vec{r_1} = -2\hat{i} + 3\hat{j} + 4\hat{k}\)
- \(\vec{r_2} = 1\hat{i} + 2\hat{j} + 3\hat{k}\)

Therefore, we calculate:
\[ \vec{r} = (1 - (-2))\hat{i} + (2 - 3)\hat{j} + (3 - 4)\hat{k} \]
Simplifying this gives:
\[ \vec{r} = (1 + 2)\hat{i} + (2 - 3)\hat{j} + (3 - 4)\hat{k} = 3\hat{i} - 1\hat{j} - 1\hat{k} \]

3. **Define the Force Vector \(\vec{F}\)**:
The force vector is given as:
\[ \vec{F} = \hat{i} + \hat{j} + \hat{k} \]

4. **Set Up the Cross Product**:
The cross product \(\vec{r} \times \vec{F}\) can be calculated using the determinant of a matrix:
\[ \vec{r} \times \vec{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -1 & -1 \\ 1 & 1 & 1 \end{vmatrix} \]

5. **Calculate the Determinant**:
Expanding the determinant:
\[ \vec{r} \times \vec{F} = \hat{i} \begin{vmatrix} -1 & -1 \\ 1 & 1 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & -1 \\ 1 & 1 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & -1 \\ 1 & 1 \end{vmatrix} \]

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

6. **Combine the Results**:
Putting it all together:
\[ \vec{r} \times \vec{F} = 0\hat{i} - 4\hat{j} + 4\hat{k} = -4\hat{j} + 4\hat{k} \]

7. **Final Result**:
Thus, the cross product \(\vec{r} \times \vec{F}\) is:
\[ \vec{r} \times \vec{F} = 4\hat{k} - 4\hat{j} \]