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 $\mathbf{u} \times \mathbf{v}$

\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.