find the area of the parralelagram that has adjacent sides u = i - 2j +2k and v = 3j - 2k

To find the area of the parallelogram formed by two vectors **u** and **v**, we use the formula for the area of a parallelogram in 3D space, which is given by the magnitude of the cross product of the two vectors:

\[
\text{Area} = ||\mathbf{u} \times \mathbf{v}||
\]

Given:

\[
\mathbf{u} = \hat{i} - 2\hat{j} + 2\hat{k}
\]
\[
\mathbf{v} = 3\hat{j} - 2\hat{k}
\]

### Step 1: Compute the cross product \(\mathbf{u} \times \mathbf{v}\)
\[
\mathbf{u} \times \mathbf{v} = 
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
1 & -2 & 2 \\
0 & 3 & -2
\end{vmatrix}
\]

Expanding the determinant:

\[
\mathbf{u} \times \mathbf{v} = \hat{i} \left( (-2)(-2) - (2)(3) \right) - \hat{j} \left( (1)(-2) - (2)(0) \right) + \hat{k} \left( (1)(3) - (-2)(0) \right)
\]

Simplifying each term:

\[
\mathbf{u} \times \mathbf{v} = \hat{i} (4 - 6) - \hat{j} (-2 - 0) + \hat{k} (3 - 0)
\]
\[
\mathbf{u} \times \mathbf{v} = -2\hat{i} + 2\hat{j} + 3\hat{k}
\]

So, the cross product is:
\[
\mathbf{u} \times \mathbf{v} = -2\hat{i} + 2\hat{j} + 3\hat{k}
\]

### Step 2: Find the magnitude of the cross product
\[
||\mathbf{u} \times \mathbf{v}|| = \sqrt{(-2)^2 + 2^2 + 3^2}
\]
\[
||\mathbf{u} \times \mathbf{v}|| = \sqrt{4 + 4 + 9} = \sqrt{17}
\]

Thus, the area of the parallelogram is:
\[
\boxed{\sqrt{17}}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How is the cross product geometrically interpreted?
2. What is the significance of the magnitude of a vector?
3. How do you compute the area of a triangle using vectors?
4. How can you find the volume of a parallelepiped using vectors?
5. What are some applications of vector cross products in physics?

**Tip**: The cross product of two vectors is always perpendicular to both vectors, making it useful for finding areas and directions in 3D.

find the area of the parallelogram that has adjacent sides u = i - 2j + 2k and v = 3j - 2k

This solution explains how to calculate the area of a parallelogram formed by vectors u = i - 2j + 2k and v = 3j - 2k using the cross product method. Understand vector algebra and learn how to apply the magnitude of cross products in 3D geometry. Suitable for high school students (Grades 10-12).

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