Given two vectors

u =

3

2

and v =

1

4

,

find the area of the parallelogram formed by these vectors using the determinant.

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.

Given two vectors u = [3, 2] and v = [1, 4], find the area of the parallelogram formed by these vectors using the determinant.

Learn how to calculate the area of a parallelogram formed by vectors [3, 2] and [1, 4] using the determinant of a 2x2 matrix. This problem involves linear algebra concepts and determinant calculations, suitable for Grades 10-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation