Problem 62 - Advanced Engineering Mathematics

Question

Prove \(\mathbf{a} \times(\mathbf{b} \times \mathbf{c})+\mathbf{b} \times(\mathbf{c} \times \mathbf{a})+\mathbf{c} \times(\mathbf{a} \times \mathbf{b})=\mathbf{0}\)

Step-by-Step Solution

Answer

The given vector identity holds true as all terms cancel to zero.

Step 1: Understanding the Vector Triple Product

The vector triple product identity states that for any three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), the expression \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \) can be expanded using the formula \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \). This is known as the vector triple product expansion or Lagrange's vector identity.

Step 2: Apply the Identity to Each Term

We will apply the vector triple product identity to each term in the expression. For \( \mathbf{a} \times(\mathbf{b} \times \mathbf{c}) \), using the identity gives us: \[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}.\]Similarly, for \( \mathbf{b} \times(\mathbf{c} \times \mathbf{a}) \) we have: \[\mathbf{b} \times (\mathbf{c} \times \mathbf{a}) = (\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}.\]And for \( \mathbf{c} \times(\mathbf{a} \times \mathbf{b}) \), it results in: \[\mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = (\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b}.\]

Step 3: Combine and Simplify the Expressions

After applying the vector triple product identity, substitute each expression back into the original equation:\[\begin{align*}&((\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}) \+ &((\mathbf{b} \cdot \mathbf{a}) \mathbf{c} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a}) \+ &((\mathbf{c} \cdot \mathbf{b}) \mathbf{a} - (\mathbf{c} \cdot \mathbf{a}) \mathbf{b}).\end{align*}\]Notice how terms can be paired:- \((\mathbf{a} \cdot \mathbf{c}) \mathbf{b}\) and \((-\mathbf{c} \cdot \mathbf{a}) \mathbf{b} \) cancel each other.- \((\mathbf{b} \cdot \mathbf{a}) \mathbf{c}\) and \((-\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \) cancel each other.- \((\mathbf{c} \cdot \mathbf{b}) \mathbf{a}\) and \((-\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \) cancel each other.

Step 4: Conclusion and Final Result

After cancelling the terms, the entire expression simplifies to zero. Thus, we have shown that\[\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) + \mathbf{b} \times (\mathbf{c} \times \mathbf{a}) + \mathbf{c} \times (\mathbf{a} \times \mathbf{b}) = \mathbf{0}.\]This demonstrates the identity and verifies the equation is correct.

Key Concepts

Lagrange's Vector Identity

Lagrange's Vector Identity is a fundamental concept in vector calculus and describes a relationship between the cross product of three vectors. This identity helps us expand expressions involving a double cross product, making them easier to understand and work with. The identity is expressed as:\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]This reveals that a double cross product can be expressed as a combination of dot and cross products involving the original vectors.
Lagrange's Vector Identity simplifies complex vector expressions, allowing within the fields of physics and engineering to solve problems related to rotational dynamics and electromagnetic fields.

Cross Product

The cross product is a mathematical operation between two vectors in three-dimensional space, resulting in a new vector that is perpendicular to both input vectors. When you take the cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \), the magnitude of the resulting vector is given by:\[ \|\mathbf{a} \times \mathbf{b}\| = \|\mathbf{a}\| \|\mathbf{b}\| \sin \theta \]Where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). This operation is useful in determining areas of parallelograms and the torque applied about an axis.

• The direction of the cross product follows the right-hand rule. If you point your index finger in the direction of \( \mathbf{a} \) and your middle finger in the direction of \( \mathbf{b} \), your thumb will point in the direction of \( \mathbf{a} \times \mathbf{b} \).
• The cross product is anti-commutative, meaning \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \).

Vector Calculus

Vector calculus is the branch of mathematics focusing on vector fields and operations in multi-dimensional spaces. It combines the use of vectors, multivariable functions, and calculus operations, like derivations and integrations. Understanding vector calculus is essential for anyone studying physical sciences or engineering, as it provides the tools to model real-world phenomena.
Key operations include:

• Gradient (\( abla f \)): Measures the rate of change of a scalar field.
• Divergence (\( abla \cdot \mathbf{F} \)): Indicates the net flow of a vector field's density out of an infinitesimal volume.
• Curl (\( abla \times \mathbf{F} \)): Represents the rotation of a vector field.

These operations, along with vector identities like Lagrange's vector identity, help in structuring and solving complex vector equations used in fluid dynamics, electromagnetism, and other advanced fields.