Find the area of the parallelogram that has **u** and **v** as adjacent sides.

**u** = −**i** + 2**j** − 2**k**

**v** = **i** + **k**
 
 square units

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.

Learn how to find the area of a parallelogram formed by vectors u and v using the cross product method. Detailed step-by-step solution with cross product formula and magnitude calculation. Suitable for advanced high school students.

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