To find the area of the parallelogram formed by vectors $\mathbf{u}$ and $\mathbf{v}$, we need to compute the magnitude of their cross product.

Given vectors: $\mathbf{u} = -\mathbf{i} + 2\mathbf{j} - 2\mathbf{k} = \begin{pmatrix} -1 \\ 2 \\ -2 \end{pmatrix}$ $\mathbf{v} = \mathbf{i} + \mathbf{k} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}$

First, compute the cross product $\mathbf{u} \times \mathbf{v}$: $\mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 2 & -2 \\ 1 & 0 & 1 \end{vmatrix}$

Expanding this determinant: $\mathbf{u} \times \mathbf{v} = \mathbf{i} \left( 2 \cdot 1 - (-2) \cdot 0 \right) - \mathbf{j} \left( -1 \cdot 1 - (-2) \cdot 1 \right) + \mathbf{k} \left( -1 \cdot 0 - 2 \cdot 1 \right)$

Simplify each component: $\mathbf{i} (2 - 0) - \mathbf{j} (-1 + 2) + \mathbf{k} (0 - 2)$ $= \mathbf{i} (2) - \mathbf{j} (1) + \mathbf{k} (-2)$

Thus, the cross product is: $\mathbf{u} \times \mathbf{v} = \begin{pmatrix} 2 \\ -1 \\ -2 \end{pmatrix}$

Next, find the magnitude of this vector to determine the area of the parallelogram: $\left| \mathbf{u} \times \mathbf{v} \right| = \sqrt{2^2 + (-1)^2 + (-2)^2}$ $= \sqrt{4 + 1 + 4}$ $= \sqrt{9}$ $= 3$

So, the area of the parallelogram is: $3 \text{ square units}$

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 area of a parallelogram in two dimensions?
2. How do you determine if two vectors are parallel?
3. What is the geometric meaning of the determinant of a matrix in two or three dimensions?
4. How can you find the volume of a parallelepiped using vectors?
5. What is the triple product of vectors, and how is it related to volume?
6. How do you normalize a vector, and why is it useful?
7. What is the significance of the vector projection in physical applications?
8. How do you use vector addition to solve problems involving forces?

Tip: Remember, the cross product's direction follows the right-hand rule, which can help visualize the perpendicular vector in three-dimensional space.