Problem 51 - Advanced Engineering Mathematics

Question

Find the volume of the parallelepiped for which the given vectors are three edges. $$ \mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j}, \mathbf{c}=2 \mathbf{i}+2 \mathbf{j}+2 \mathbf{k} $$

Step-by-Step Solution

Answer

The volume of the parallelepiped is 10.

Step 1: Represent Vectors as Columns

Represent vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) as columns in a 3x3 matrix. This is the initial setup: \[\mathbf{a} = \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix},\mathbf{b} = \begin{pmatrix} -1 \ 4 \ 0 \end{pmatrix},\mathbf{c} = \begin{pmatrix} 2 \ 2 \ 2 \end{pmatrix}\]

Step 2: Construct the Matrix

The matrix formed by the vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) is as follows: \[\begin{bmatrix} 1 & -1 & 2 \ 1 & 4 & 2 \ 0 & 0 & 2 \end{bmatrix}\]

Step 3: Calculate the Determinant

Calculate the determinant of the matrix to find the volume. The determinant is calculated as follows: \[|\begin{vmatrix} 1 & -1 & 2 \ 1 & 4 & 2 \ 0 & 0 & 2 \end{vmatrix}| \]Using cofactor expansion along the third column: \[= 2 \left( \begin{vmatrix} 1 & -1 \ 1 & 4 \end{vmatrix} \right) = 2(4 + 1) = 2 \times 5 = 10\]

Step 4: Interpret the Result

The determinant calculated is 10, which represents the volume of the parallelepiped formed by vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \).

Key Concepts

Vector Calculus

Vector Calculus is a branch of mathematics that deals with vectors and operations on vector spaces. It helps us understand physical quantities like force and velocity, which have both magnitude and direction.

• Vectors: Basic building blocks in vector calculus. They are entities with both direction and magnitude represented in Euclidean space.
• Applications: Useful in physics and engineering, such as calculating trajectories, fluid dynamics, and electromagnetism.

We use vector calculus to solve various geometric and physical problems, like calculating the volume of a parallelepiped formed by three vectors in space. The concept involves various mathematical tools including dot products, cross products, and vector operations. Here, the determinant of a matrix formed by three vectors gives the volume.

Determinants

Determinants play a pivotal role in various mathematical computations. They are crucial in linear algebra for solving systems of linear equations and analyzing matrix properties.

• Matrix Requirement: Determinants are computed for square matrices (matrices with the same number of rows and columns).
• Properties of Determinants: They can provide information on the volume scaling factor of linear transformations.
• Significance: A zero determinant indicates a matrix that doesn't have an inverse, representing parallel or dependent vectors in space.

In our exercise, we compute the determinant of a 3x3 matrix to find the volume of the parallelepiped, reflecting how matrix transformations affect space.

Matrix Representation

Matrices are a fundamental part of mathematics and are used for numerous calculations in vector calculus. They compactly organize data, making it easier to perform complex operations.

• Column Vectors: Each vector in a 3-dimensional space can be represented as a column in a matrix, preserving its structure.
• Matrix Formation: By arranging vectors as columns, we can visualize geometric transformations and solve for volumes through determinant calculations.

Here, the vectors.\[\mathbf{a} = \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}, \mathbf{b} = \begin{pmatrix} -1 \ 4 \ 0 \end{pmatrix}, \mathbf{c} = \begin{pmatrix} 2 \ 2 \ 2 \end{pmatrix}\] are placed into a matrix allowing us to compute its determinant for the parallelepiped's volume.

Cofactor Expansion

Cofactor Expansion, also known as Laplace Expansion, is a method used to calculate the determinant of a matrix. It involves breaking down a matrix into smaller parts, making it easier to manage calculations for larger matrices.

• Usefulness: Particularly useful for calculating determinants of 3x3 or larger matrices by expanding along a row or column.
• Steps: Select a row or column, remove it to form a smaller matrix, calculate its determinant, and multiply by the signed cofactor.
• Alternating Signs: Remember that the cofactors alternate signs (+/-) based on their position.

In our specific solution, we expanded along the third column of the matrix to easily compute its determinant and subsequently, the volume of the parallelepiped.