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To solve the problem of finding the vector moment of the force \(\vec{F} = 4\hat{i} - 3\hat{j}\) acting at point \(A(2, 3)\) with respect to point \(B(-1, 2)\), we need to use the concept of **torque (moment of a force)**, which is given by:

\[
\vec{M} = \vec{r} \times \vec{F}
\]

Where:
- \(\vec{M}\) is the moment vector,
- \(\vec{r}\) is the position vector from point \(B\) to point \(A\),
- \(\vec{F}\) is the force vector,
- \(\times\) denotes the cross product.

### Step 1: Find the position vector \(\vec{r}\)
The position vector \(\vec{r}\) from point \(B(-1, 2)\) to point \(A(2, 3)\) is:

\[
\vec{r} = \vec{A} - \vec{B} = (2 - (-1))\hat{i} + (3 - 2)\hat{j} = 3\hat{i} + 1\hat{j}
\]

### Step 2: Set up the cross product
Now we compute the cross product \(\vec{r} \times \vec{F}\), where:

\[
\vec{r} = 3\hat{i} + 1\hat{j}, \quad \vec{F} = 4\hat{i} - 3\hat{j}
\]

The cross product in two dimensions gives a vector in the \(\hat{k}\)-direction:

\[
\vec{M} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
3 & 1 & 0 \\
4 & -3 & 0
\end{vmatrix}
\]

Expanding the determinant:

\[
\vec{M} = \hat{k}(3 \times (-3) - 1 \times 4) = \hat{k}(-9 - 4) = \hat{k}(-13)
\]

### Step 3: Final result
Thus, the moment of the force is:

\[
\vec{M} = -13\hat{k}
\]

The correct answer is therefore:

\[
\boxed{a} \ -13\hat{k}
\]

---

Would you like further details on any part of the process, or do you have any other questions? Here are 5 related questions to explore:

1. How is the cross product of vectors calculated in three dimensions?
2. What are the physical implications of the vector moment being in the \(\hat{k}\)-direction?
3. How would the moment change if the force had a \(\hat{k}\)-component as well?
4. What are some common applications of torque in mechanical systems?
5. How does the moment arm affect the magnitude of torque?

**Tip:** In problems involving torque, always make sure to properly define the position vector relative to the point about which you're calculating the moment.

If the force F = 4i - 3j acts at the point A(2, 3), then the vector moment of this force with respect to the point B(-1, 2) equals?

Learn how to calculate the vector moment (torque) of a force F = 4i - 3j acting at point A(2, 3) with respect to point B(-1, 2). This problem explores vector algebra concepts such as position vectors, cross products, and determinant methods. Ideal for college-level physics and engineering students.

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