Problem 32 - Thomas Calculus

Question

Cross products of three vectors Show that except in degenerate cases, \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\) lies in the plane of \(\mathbf{u}\) and \(\mathbf{v},\) whereas \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})\) lies in the plane of \(\mathbf{v}\) and \(\mathbf{w} .\) What are the degenerate cases?

Step-by-Step Solution

Answer

The expressions lie in their respective planes; degenerate cases involve zero or parallel vectors.

Step 1: Understand the Vector Triple Product Identity

The vector triple product identity states that for three vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \), the expression \( (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \). This means that the result of the vector triple product is a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).

Step 2: Apply the Identity to \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\)

Applying the identity to \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\), we get:\[(\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = ((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}) \mathbf{v} - (\mathbf{v} \cdot \mathbf{w}) (\mathbf{u} \times \mathbf{v})\]This expression is a linear combination of \(\mathbf{v}\) and \(\mathbf{u} \times \mathbf{v}\), which means it lies in the plane defined by \( \mathbf{u} \) and \( \mathbf{v} \).

Step 3: Apply the Identity to \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w})\)

Applying the identity to \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w})\), we have:\[\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \cdot \mathbf{w}) \mathbf{v} - (\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\]This expression is a linear combination of \(\mathbf{v}\) and \(\mathbf{w}\), meaning it lies in the plane defined by \(\mathbf{v}\) and \(\mathbf{w}\).

Step 4: Identify Degenerate Cases

Degenerate cases occur when the plane defined by the vectors doesn't exist, i.e., when the vectors are linearly dependent. In such cases, \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \) are either parallel or one of the vectors is the zero vector, leading to zero cross-products and thus zero results for the triple products.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector algebra, crucial for calculating various physical quantities. It gives a vector that is perpendicular to two given vectors in three-dimensional space.
The symbol for cross product between two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is \( \mathbf{a} \times \mathbf{b} \). This operation differs from the dot product, which results in a scalar value.
Partial formulas related to the cross product are:

• The magnitude of \( \mathbf{a} \times \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin\theta \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \).
• The direction is given by the right-hand rule, where the thumb points in the direction of the result.

The cross product is essential in situations needing perpendicularity, such as determining the normal vector to a plane.

Linear Dependence

Linear dependence is a concept involving vectors and whether they can be represented as a combination of others. If vectors are linearly dependent, one vector can be a scalar multiple or a linear combination of the others.
For vectors \( \mathbf{a}, \mathbf{b}, \) and \( \mathbf{c} \), they are linearly dependent if there are constants, not all zero, such that: \[ c_1 \mathbf{a} + c_2 \mathbf{b} + c_3 \mathbf{c} = \mathbf{0} \]This scenario implies that the vectors lie on the same plane or line.
This concept is related to when certain expressions, like the vector triple product, lead to zero, indicating dependency among the vectors, such as when planes are degenerate cases.

Degenerate Cases

Degenerate cases are situations where typical geometric or algebraic configurations do not hold. In the context of the vector triple product, it refers to scenarios when two or more of the involved vectors are linearly dependent.
This occurs in cases like:

• All vectors being parallel, meaning they lie on the same line, eliminate all perpendicularly oriented vectors.
• One vector being the zero vector, hence contributing nothing to cross products.

These cases mean there is no defined plane for the vectors, typically reducing expressions to redundant or zero-valued results, serving as checks for dependency and structure in vector analysis.

Planes Defined by Vectors

Vectors play a crucial role in defining planes in three-dimensional space. A plane can be characterized using two non-parallel vectors that span it.
Mathematically, a plane containing vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed as the set of all points \( \mathbf{p} \) given by: \[ \mathbf{p} = \mathbf{r} + s \mathbf{a} + t \mathbf{b} \]where \( \mathbf{r} \) is a point on the plane, and \( s, t \) are scalars.
The cross product \( \mathbf{a} \times \mathbf{b} \) gives a normal vector, perpendicular to this plane. Understanding how vectors define planes is key in vector operations, especially when dealing with the vector triple product and related vector algebra concepts.