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.