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The problem is asking for the area of a triangle \( PQR \) given the coordinates of its vertices:

- \( P = (4, 4, 2) \)
- \( Q = (-6, -3, -4) \)
- \( R = (-1, 1, 4) \)

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

\[
\text{Area of triangle} = \frac{1}{2} \left| \mathbf{PQ} \times \mathbf{PR} \right|
\]

where:
- \( \mathbf{PQ} \) is the vector from \( P \) to \( Q \) (i.e., \( \mathbf{PQ} = Q - P \))
- \( \mathbf{PR} \) is the vector from \( P \) to \( R \) (i.e., \( \mathbf{PR} = R - P \))
- \( \times \) represents the cross product of two vectors
- \( | \cdot | \) represents the magnitude of the resulting vector.

Let me compute the area for you.The area of the triangle \( PQR \) is approximately \( 29.79 \) square units.

Would you like a detailed explanation of the steps involved in the calculation or have any questions?

Here are 5 related questions you might find interesting:
1. How do you calculate the cross product of two vectors?
2. What is the significance of the magnitude of a cross product?
3. How do you find the distance between two points in 3D space?
4. Can this method be applied to find the area of a quadrilateral in 3D?
5. How does the area formula for a triangle in 2D differ from that in 3D?

**Tip:** When dealing with vector problems, always ensure that you understand the geometric meaning of the operations (like dot products and cross products) in addition to just performing the calculations.

Find the area of a triangle PQR, where P = (4, 4, 2), Q = (-6, -3, -4), and R = (-1, 1, 4).

This solution explains how to find the area of a triangle in 3D space with vertices at P(4,4,2), Q(-6,-3,-4), and R(-1,1,4) using vector cross products. Detailed steps cover how to calculate vectors PQ and PR, compute their cross product, and determine the magnitude for the area. Suitable for high school geometry and vector algebra students.

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