W1 3. Let w = V = 2 W2 gram formed by w, v is equal to 2.5 1.5 0.5 is W Referring to the image below, show that the area of the parallelo- |w102 - 20₁| = |det (₂2)| W2 V2 2.5
W1 3. Let w = V = 2 W2 gram formed by w, v is equal to 2.5 1.5 0.5 is W Referring to the image below, show that the area of the parallelo- |w102 - 20₁| = |det (₂2)| W2 V2 2.5
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Linear Algebra: Area of a Parallelogram
**Problem Statement:**
Let \( \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \). Referring to the image below, show that the area of the parallelogram formed by \( \mathbf{w} \) and \( \mathbf{v} \) is equal to
\[ |w_1 v_2 - w_2 v_1| = \left| \text{det} \begin{pmatrix} w_1 & v_1 \\ w_2 & v_2 \end{pmatrix} \right|. \]
**Diagram Explanation:**
The diagram provided depicts a parallelogram formed by the vectors \( \mathbf{w} \) and \( \mathbf{v} \) in a coordinate system.
- The x-axis ranges from 0 to approximately 4.
- The y-axis ranges from 0 to approximately 3.5.
- One vertex of the parallelogram is at the origin (0, 0).
- The vector \( \mathbf{w} \) has its tail at the origin and its head at coordinates corresponding to \( \mathbf{w} \).
- Similarly, the vector \( \mathbf{v} \) has its tail at the origin and its head at coordinates corresponding to \( \mathbf{v} \).
- Two additional vertices of the parallelogram are located where the tip of \( \mathbf{w} \) is added to \( \mathbf{v} \), and the tip of \( \mathbf{v} \) is added to \( \mathbf{w} \).
### Analytical Explanation:
To calculate the area of the parallelogram defined by \( \mathbf{w} \) and \( \mathbf{v} \):
1. **Vectors:** Represent the vectors in 2D space as
\[
\mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}.
\]
2. **Determinant:** The area \( A \) of](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0761ca0b-8a30-47ed-b77e-00687bf0302e%2F90a4c294-b5be-42fa-941f-c75f335be974%2Fzvxwgn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra: Area of a Parallelogram
**Problem Statement:**
Let \( \mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix} \) and \( \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix} \). Referring to the image below, show that the area of the parallelogram formed by \( \mathbf{w} \) and \( \mathbf{v} \) is equal to
\[ |w_1 v_2 - w_2 v_1| = \left| \text{det} \begin{pmatrix} w_1 & v_1 \\ w_2 & v_2 \end{pmatrix} \right|. \]
**Diagram Explanation:**
The diagram provided depicts a parallelogram formed by the vectors \( \mathbf{w} \) and \( \mathbf{v} \) in a coordinate system.
- The x-axis ranges from 0 to approximately 4.
- The y-axis ranges from 0 to approximately 3.5.
- One vertex of the parallelogram is at the origin (0, 0).
- The vector \( \mathbf{w} \) has its tail at the origin and its head at coordinates corresponding to \( \mathbf{w} \).
- Similarly, the vector \( \mathbf{v} \) has its tail at the origin and its head at coordinates corresponding to \( \mathbf{v} \).
- Two additional vertices of the parallelogram are located where the tip of \( \mathbf{w} \) is added to \( \mathbf{v} \), and the tip of \( \mathbf{v} \) is added to \( \mathbf{w} \).
### Analytical Explanation:
To calculate the area of the parallelogram defined by \( \mathbf{w} \) and \( \mathbf{v} \):
1. **Vectors:** Represent the vectors in 2D space as
\[
\mathbf{w} = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}, \quad \mathbf{v} = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}.
\]
2. **Determinant:** The area \( A \) of
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