Question

Prove the Jacobi identity: \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})+\mathbf{v} \times(\mathbf{w} \times \mathbf{u})+\mathbf{w} \times(\mathbf{u} \times \mathbf{v})=\mathbf{0}\).

Short answer

The Jacobi identity is proven using vector triple product identity and simplification.

Step-by-step solution

Understand the Cross Product

Before attempting to prove the Jacobi identity, it's essential to understand the properties of the cross product of vectors. The cross product is a binary operation on two vectors in three-dimensional space and is denoted by \( \mathbf{a} \times \mathbf{b} \). The result is a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).

Review the Vector Triple Product Identity

The vector triple product identity is a useful tool in proving the Jacobi identity. It states that for three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), the cross product \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \). This identity will be applied repeatedly in the proof.

Apply the Vector Triple Product Identity to Each Term

Apply the vector triple product identity to each term in the Jacobi identity:1. \( \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{u} \cdot \mathbf{v})\mathbf{w} \).2. \( \mathbf{v} \times (\mathbf{w} \times \mathbf{u}) = (\mathbf{v} \cdot \mathbf{u})\mathbf{w} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u} \).3. \( \mathbf{w} \times (\mathbf{u} \times \mathbf{v}) = (\mathbf{w} \cdot \mathbf{v})\mathbf{u} - (\mathbf{w} \cdot \mathbf{u})\mathbf{v} \).

Sum the Results of the Triple Products

Now, substitute the expressions obtained in Step 3 back into the Jacobi identity and sum them:- Substitute the expressions: \((\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{u} \cdot \mathbf{v})\mathbf{w} + (\mathbf{v} \cdot \mathbf{u})\mathbf{w} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u} + (\mathbf{w} \cdot \mathbf{v})\mathbf{u} - (\mathbf{w} \cdot \mathbf{u})\mathbf{v}\).- Group similar terms based on the vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \).

Simplify and Prove the Identity

Group and simplify the terms from Step 4:1. \((\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{w} \cdot \mathbf{u})\mathbf{v} = 0\)2. \(-(\mathbf{u} \cdot \mathbf{v})\mathbf{w} + (\mathbf{v} \cdot \mathbf{u})\mathbf{w} = 0\)3. \((\mathbf{w} \cdot \mathbf{v})\mathbf{u} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u} = 0\)Each pair results in zero when grouped appropriately, confirming that the left side of Jacobi identity simplifies to \( \mathbf{0} \).

Conclusion: Prove the Jacobi Identity

By using the vector triple product identity and simplifying each term, we've shown that the Jacobi identity \( \mathbf{u} \times (\mathbf{v} \times \mathbf{w}) + \mathbf{v} \times (\mathbf{w} \times \mathbf{u}) + \mathbf{w} \times (\mathbf{u} \times \mathbf{v}) = \mathbf{0} \) holds true.

Key concepts

Vector Cross Product

The vector cross product is a critical concept in vector algebra, particularly when dealing with three-dimensional vectors. It is denoted as \( \mathbf{a} \times \mathbf{b} \) and results in a new vector that is perpendicular to the plane formed by the original vectors \( \mathbf{a} \) and \( \mathbf{b} \). This operation is significant because it allows us to calculate a vector that is orthogonal to two given vectors, hence providing useful insights into their spatial relationship.
The direction of the resulting vector from the cross product adheres to the right-hand rule: If you point your thumb in the direction of \( \mathbf{a} \) and your index finger in the direction of \( \mathbf{b} \), then the resulting vector \( \mathbf{a} \times \mathbf{b} \) points in the direction of your middle finger.
Moreover, the magnitude of this vector is given by the formula:
\[ |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| |\mathbf{b}| \sin(\theta) \]
where \( \theta \) is the angle between the two vectors. This property is particularly useful when dealing with physical quantities like torque, angular momentum, where direction and magnitude are paramount.

Vector Triple Product Identity

The vector triple product identity is a powerful tool in vector algebra. It simplifies the process of dealing with nested cross products, which is crucial in proofs and geometric applications. The identity states:
\[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]
This formula rewrites a cross product of a cross product into a linear combination of vectors. It's significant because it allows for the simplification of expressions that initially seem complex.
In the context of solving the Jacobi identity, this identity is repeatedly applied to express and later simplify terms. It transforms complex cross-product terms into manageable dot product terms which are easier to group and simplify.
Understanding and applying the vector triple product identity is essential for students working with higher-level vector calculations, allowing them to transform three-dimensional problems into more intuitive scalar formulations.

Vector Algebra

Vector algebra is a branch of mathematics that handles the addition and multiplication of vectors, which are quantities having both a magnitude and a direction. It provides vital tools for understanding and manipulating physical quantities in engineering and physics.
The essential operations in vector algebra include:

• Vector addition: Combining vectors to determine their resultant.
• Scalar multiplication: Stretching or shrinking vectors by a scalar.
• Dot product: Producing a scalar from two vectors, representing the cosine of their angle.
• Cross product: Creating a vector from two vectors, incorporating their mutual perpendicularity.

Vector algebra allows us to better understand the spatial relationships between physical entities. It simplifies calculations related to force, velocity, and other directional quantities, ensuring that we can effectively interpret and manipulate the properties and interactions of objects in three-dimensional space.

Proof Techniques in Mathematics

Proof techniques are the foundational tools of mathematics, crucial for validating claims and establishing truths in mathematical theories and problems. They are methods used to demonstrate the logical validity of mathematical statements.
In the context of the Jacobi identity in vector algebra, the proof involves several techniques:

• Direct proof: Applying known identities like the vector triple product directly to verify the statement.
• Algebraic manipulation: Rearranging and simplifying expressions using properties of operations such as commutativity, distributivity, and associativity.
• Case analysis: Breaking down complex problems into manageable parts and proving each part independently.

Each technique is key in building a step-by-step argument that ultimately leads to the desired conclusion.
Familiarity with proof techniques not only enhances one's ability to solve mathematical problems but also improves critical thinking and logic skills that are applicable in many real-world scenarios.