To find the area of a parallelogram formed by two vectors $\mathbf{u}$ and $\mathbf{v}$ in three-dimensional space, you can use the cross product of the vectors. The magnitude of the cross product gives the area of the parallelogram.

Given vectors: $\mathbf{u} = \begin{pmatrix} 4 \\ 4 \\ -2 \end{pmatrix}$ $\mathbf{v} = \begin{pmatrix} 0 \\ 4 \\ 2 \end{pmatrix}$

First, find the cross product $\mathbf{u} \times \mathbf{v}$: $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 4 & -2 \\ 0 & 4 & 2 \end{vmatrix}$

Expanding this determinant, we get: $\mathbf{u} \times \mathbf{v} = \mathbf{i} \left( 4 \cdot 2 - (-2) \cdot 4 \right) - \mathbf{j} \left( 4 \cdot 2 - (-2) \cdot 0 \right) + \mathbf{k} \left( 4 \cdot 4 - 4 \cdot 0 \right)$

Simplify each component: $\mathbf{i} (8 + 8) - \mathbf{j} (8 - 0) + \mathbf{k} (16 - 0)$ $= \mathbf{i} (16) - \mathbf{j} (8) + \mathbf{k} (16)$

Thus, the cross product is: $\mathbf{u} \times \mathbf{v} = \begin{pmatrix} 16 \\ -8 \\ 16 \end{pmatrix}$

Next, find the magnitude of this vector to determine the area of the parallelogram: $\left| \mathbf{u} \times \mathbf{v} \right| = \sqrt{16^2 + (-8)^2 + 16^2}$ $= \sqrt{256 + 64 + 256}$ $= \sqrt{576}$ $= 24$

So, the area of the parallelogram is: $24 \text{ square units}$

Would you like any further details or have any questions?

Here are 8 related questions you might find interesting:

1. How do you find the volume of a parallelepiped formed by three vectors?
2. What is the geometric significance of the dot product of two vectors?
3. How can you determine if two vectors are perpendicular?
4. What is the formula for the area of a triangle formed by two vectors?
5. How do you calculate the angle between two vectors in space?
6. What is the projection of one vector onto another?
7. How do you find the vector component of one vector along another?
8. How can you use vector operations to solve real-world problems?

Tip: The cross product is only defined in three dimensions and results in a vector that is perpendicular to the plane formed by the original vectors.