Math Problem Statement
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
Solution
The volume of the parallelepiped defined by the three linearly independent vectors , , and 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 :
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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Vectors
Linear Independence
Determinants
Volume Calculation
Formulas
Scalar triple product
Determinant formula for 3x3 matrix
Theorems
-
Suitable Grade Level
Grades 11-12
Related Recommendation
Calculate Volumes: Parallelepiped and Tetrahedron - Detailed Solution
Volume of Parallelepiped Induced by Vectors Using Determinants
Calculate Volume of Parallelepiped from Given Vertices
How to Find the Volume of a Parallelepiped with Given Vectors
Volume of a Parallelepiped Using Vectors π’β, π£, and π€β