Problem 30 - Thomas Calculus

Question

Compute \((\mathbf{i} \times \mathbf{j}) \times \mathbf{j}\) and \(\mathbf{i} \times(\mathbf{j} \times \mathbf{j}) .\) What can you conclude about the associativity of the cross product?

Step-by-Step Solution

Answer

The cross product is not associative.

Step 1: Understand the Cross Product

The cross product \( \mathbf{u} \times \mathbf{v} \) results in a vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \). For the standard unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \), we have \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \).

Step 2: Calculate \((\mathbf{i} \times \mathbf{j}) \times \mathbf{j}\)

First, calculate \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \). Next, compute \( \mathbf{k} \times \mathbf{j} \). Since \( \mathbf{k}, \mathbf{j}, \) and \( \mathbf{i} \) are cyclic permutations of \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), \( \mathbf{k} \times \mathbf{j} = -\mathbf{i} \) according to the right-hand rule.

Step 3: Calculate \(\mathbf{i} \times (\mathbf{j} \times \mathbf{j})\)

Since the cross product of any vector with itself is zero, \( \mathbf{j} \times \mathbf{j} = \mathbf{0} \). This leads to \( \mathbf{i} \times \mathbf{0} = \mathbf{0} \).

Step 4: Compare Results and Conclusions on Associativity

We obtained the results \((\mathbf{i} \times \mathbf{j}) \times \mathbf{j} = -\mathbf{i}\) and \(\mathbf{i} \times (\mathbf{j} \times \mathbf{j}) = \mathbf{0}\). Since these results are different, it demonstrates that the cross product is not associative.

Key Concepts

Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities having both direction and magnitude. Vectors are essential in physics and engineering, being utilized to represent physical quantities such as force and velocity. Unlike scalar quantities, vectors require both a magnitude and a direction for a complete description.

• Components: A vector can be broken into components along the coordinate axes, typically represented as unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).
• Operations: Operations such as addition, subtraction, and multiplication (dot and cross products) are used to analyze and manipulate vectors.

Cross product is one way to multiply two vectors resulting in another vector, following specific algebraic rules and geometric interpretations.

Associativity

Associativity refers to the property of an operation where the grouping of operands does not change the result. However, the cross product is an operation that does not adhere to this property. As the given exercise has shown:

• For vectors \( \mathbf{A} \) and \( \mathbf{B} \), the expression \((\mathbf{A} \times \mathbf{B}) \times \mathbf{C}\) is not generally equal to \(\mathbf{A} \times (\mathbf{B} \times \mathbf{C})\).
• The computed outcomes of \((\mathbf{i} \times \mathbf{j}) \times \mathbf{j} = -\mathbf{i}\) and \(\mathbf{i} \times (\mathbf{j} \times \mathbf{j}) = \mathbf{0}\) show that grouping matters in cross products.

Understanding this distinction is crucial for correctly manipulating vector quantities using cross products.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are fundamental in vector algebra, often used to specify directions without concern for magnitude.

• Notation: Usually denoted as \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) in a three-dimensional Cartesian coordinate system.
• Properties: These vectors are orthogonal to each other, meaning the angle between any pair is 90 degrees, and they form the basis of 3D vector space.
• Cross Product Relations: The cross product follows specific rules with these unit vectors, such as \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \), \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \), and \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \).

Unit vectors simplify analysis and computation in vector problems by providing straightforward reference points for direction and orientation.