Consider the following two vectors in R
n
, n > 2:
v
1
:= (0, 1, 1, . . . , 1, 1)T
,
v
2
:= (1, 0, a, . . . , a, a)
T
,
where a is an arbitrary number. Compute the area of the parallelogram generated
by the vectors v
1 and v
2
.

To compute the area of the parallelogram generated by two vectors \( v_1 \) and \( v_2 \) in \( \mathbb{R}^n \), we use the magnitude of the **cross product** of the vectors. However, in dimensions higher than \( \mathbb{R}^3 \), this is generalized by computing the **norm of the wedge product** or simply the magnitude of the determinant of the matrix formed by these vectors.

Given that \( v_1 = (0, 1, 1, \dots, 1)^T \) and \( v_2 = (1, 0, a, \dots, a)^T \), we can compute the area of the parallelogram spanned by these two vectors using the formula for the magnitude of the 2D **determinant**:

### Step 1: Express the vectors explicitly

We have:

\[
v_1 = (0, 1, 1, \dots, 1)^T
\]
and
\[
v_2 = (1, 0, a, \dots, a)^T
\]

Let \( v_1 \in \mathbb{R}^n \) where \( v_1 = (0, 1, 1, \dots, 1)^T \) consists of 1's from the second entry onwards, and \( v_2 = (1, 0, a, \dots, a)^T \) consists of 1's followed by \( a \) from the third entry onwards.

### Step 2: Compute the 2x2 determinant

The area of the parallelogram is given by the determinant of the matrix formed by \( v_1 \) and \( v_2 \) restricted to their first two components, since only the first two components contribute non-trivial entries:

\[
\text{Area} = \left| \det \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \right|
\]

The determinant of this matrix is:

\[
\det \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = -1
\]

Thus, the absolute value is \( | -1 | = 1 \).

### Step 3: Conclusion

The area of the parallelogram generated by \( v_1 \) and \( v_2 \) is \( 1 \).

### Summary:
The area of the parallelogram generated by the vectors \( v_1 \) and \( v_2 \) is \( \boxed{1} \).

Would you like further clarification or additional details?

Here are some follow-up questions for further exploration:

1. How do we compute the volume of the parallelepiped formed by three vectors?
2. How would this process change if we were dealing with vectors in \( \mathbb{R}^3 \)?
3. What is the geometric interpretation of the cross product in \( \mathbb{R}^3 \)?
4. Can the determinant method be extended to more than two vectors in higher dimensions?
5. How does the choice of the scalar \( a \) affect the area of the parallelogram?

**Tip:** The determinant gives the signed area or volume in different contexts, with the absolute value providing the magnitude regardless of orientation.

Consider the following two vectors in R^n, n > 2: v_1 := (0, 1, 1, . . . , 1, 1)^T, v_2 := (1, 0, a, . . . , a, a)^T, where a is an arbitrary number. Compute the area of the parallelogram generated by the vectors v_1 and v_2.

This problem computes the area of a parallelogram formed by two vectors in R^n using the determinant formula. Explore the key concepts in linear algebra, including vector geometry, determinants, and their application in higher dimensions. Ideal for undergraduate math 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