To find the area of the parallelogram formed by two vectors $\mathbf{u}$ and $\mathbf{v}$ using the determinant, you can use the formula:

$\text{Area} = |\det(\mathbf{A})|$

where $\mathbf{A}$ is a 2x2 matrix formed by placing $\mathbf{u}$ and $\mathbf{v}$ as columns:

$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$

Step 1: Find the Determinant

The determinant of a 2x2 matrix $\mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ is calculated as:

$\det(\mathbf{A}) = ad - bc$

For our matrix:

$\det(\mathbf{A}) = (3)(4) - (2)(1) = 12 - 2 = 10$

Step 2: Calculate the Absolute Value

The area of the parallelogram is the absolute value of the determinant:

$\text{Area} = |10| = 10$

Final Answer

The area of the parallelogram formed by the vectors $\mathbf{u}$ and $\mathbf{v}$ is 10 square units.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How do you find the area of a parallelogram in 3D space using vectors?
2. What happens to the area if one of the vectors is multiplied by a scalar?
3. How does the cross product relate to the area of a parallelogram in 3D?
4. What are some geometric interpretations of the determinant in 2D and 3D space?
5. How can you determine if two vectors are parallel using the determinant?

Tip: The absolute value of the determinant represents the area because it accounts for orientation, ensuring the area is positive regardless of the direction of the vectors.