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The problem asks to find the area of the triangle with vertices at points \((1, -1, 3)\), \((4, -3, 1)\), and \((3, -1, 2)\) in 3D space.

To find the area of a triangle in 3D space, we can use the following formula:

\[
\text{Area} = \frac{1}{2} \| \vec{AB} \times \vec{AC} \|
\]

Where:
- \(\vec{AB}\) is the vector from point A to point B.
- \(\vec{AC}\) is the vector from point A to point C.
- \(\times\) denotes the cross product, and \(\|\cdot\|\) represents the magnitude of the resulting vector.

### Step-by-step:

1. **Points:**
   - \( A(1, -1, 3) \)
   - \( B(4, -3, 1) \)
   - \( C(3, -1, 2) \)

2. **Vectors:**
   - \(\vec{AB} = B - A = (4 - 1, -3 + 1, 1 - 3) = (3, -2, -2)\)
   - \(\vec{AC} = C - A = (3 - 1, -1 + 1, 2 - 3) = (2, 0, -1)\)

3. **Cross Product** \(\vec{AB} \times \vec{AC}\):
   Using the determinant form of the cross product:
   \[
   \vec{AB} \times \vec{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & -2 & -2 \\ 2 & 0 & -1 \end{vmatrix}
   \]

   I'll now calculate this cross product.The cross product \(\vec{AB} \times \vec{AC}\) is \([2, -1, 4]\).

The magnitude of this vector is approximately 4.58, and thus the area of the triangle is:

\[
\text{Area} = \frac{1}{2} \times 4.58 \approx 2.29
\]

So, the area of the triangle is approximately \(2.29\) square units.

Would you like further clarification or details?

Here are 5 related questions you might find interesting:
1. How do you calculate the area of a triangle in 2D space?
2. What is the geometric significance of the cross product in 3D?
3. Can the formula for the area of a triangle be extended to polygons?
4. How is the magnitude of a vector calculated?
5. What are some applications of cross products in physics?

**Tip:** The cross product of two vectors is always perpendicular to the plane formed by the two vectors.

Find the area of the triangle with vertices at (1, -1, 3), (4, -3, 1), and (3, -1, 2).

This problem demonstrates how to find the area of a triangle with vertices in 3D space using vectors and cross products. Learn the step-by-step solution using geometry, vector operations, and magnitudes.

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