The volume defined by the three linearly independent vectors $\mathbf{a} = \begin{pmatrix} 1 \\ 1 \\ 1  \end{pmatrix}$,  $\mathbf{b} = \begin{pmatrix} -1 \\ 1 \\ 1  \end{pmatrix}$ and $\mathbf{c} = \begin{pmatrix} 1 \\ -1 \\ 1  \end{pmatrix}$ equals :

Select one:

a. +8

b. $\sqrt{4}$

c. 2

d. 4

e. None of the these

The volume of the parallelepiped defined by the three linearly independent vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ is given by the absolute value of the scalar triple product of these vectors, which can be computed as the determinant of the matrix formed by placing these vectors as columns.

Let's form the matrix $ A $:

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

To find the volume, we need to compute the determinant of this matrix:

\[ \det(A) = \begin{vmatrix}
1 & -1 & 1 \\
1 & 1 & -1 \\
1 & 1 & 1
\end{vmatrix} \]

We can compute this determinant using cofactor expansion along the first row:

\[
\det(A) = 1 \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix}
- (-1) \begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix}
+ 1 \begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix}
\]

Now, calculate the 2x2 determinants:

\[
\begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} = (1)(1) - (-1)(1) = 1 + 1 = 2
\]

\[
\begin{vmatrix} 1 & -1 \\ 1 & 1 \end{vmatrix} = 2 \quad (\text{same as above})
\]

\[
\begin{vmatrix} 1 & 1 \\ 1 & 1 \end{vmatrix} = (1)(1) - (1)(1) = 1 - 1 = 0
\]

Putting these values back into the expansion:

\[
\det(A) = 1 \cdot 2 - (-1) \cdot 2 + 1 \cdot 0 = 2 + 2 + 0 = 4
\]

Therefore, the volume of the parallelepiped is:

\[
|\det(A)| = |4| = 4
\]

So, the correct answer is:

d. 4

Would you like more details or have any questions?

Here are some related questions you can ask:

1. How do you compute the scalar triple product of three vectors?
2. What is the geometric interpretation of the determinant of a matrix?
3. How can the volume of a parallelepiped be visualized?
4. What is the significance of linear independence in determining volume?
5. How can cofactor expansion be used for larger matrices?

**Tip:** The volume of a parallelepiped in $\mathbb{R}^3$ can be found using the absolute value of the determinant of a matrix whose columns are the vectors defining the parallelepiped.

Learn how to calculate the volume of a parallelepiped defined by vectors using determinants. This problem involves finding the volume using the scalar triple product and determinant formulas for 3x3 matrices. Suitable for advanced high school students (Grades 11-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation