Question

Let \(\mathbf{a}\) and \(\mathbf{b}\) be nonparallel vectors, and let \(\mathbf{c}\) be any nonzero vector. Show that \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) is a vector in the plane of \(\mathbf{a}\) and \(\mathbf{b}\).

Short answer

The vector is in the plane of \(\mathbf{a}\) and \(\mathbf{b}\) because it is a linear combination of them.

Step-by-step solution

Understand the cross product property

To solve this problem, we'll use the vector triple product identity: \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\). This identity shows how to express a cross product of a cross product in terms of dot products and vectors.

Apply the vector triple product identity

Using the identity, we can substitute: \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\). This result is a linear combination of \(\mathbf{a}\) and \(\mathbf{b}\).

Recognize the vector is in the plane of \(\mathbf{a}\) and \(\mathbf{b}\)

Since \((\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\) is a linear combination of \(\mathbf{a}\) and \(\mathbf{b}\), it clearly lies in the plane spanned by \(\mathbf{a}\) and \(\mathbf{b}\). This confirms that \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) is indeed in the plane of \(\mathbf{a}\) and \(\mathbf{b}\).

Key concepts

Understanding the Cross Product

The cross product is a fundamental operation in vector spaces, especially when dealing with three-dimensional space.It is an algebraic operation that takes two input vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), and returns a third vector \( \mathbf{a} \times \mathbf{b} \).
This result is unique because the output vector is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \), following the right-hand rule.

One key characteristic of the cross product is that its magnitude equals the area of the parallelogram that has \( \mathbf{a} \) and \( \mathbf{b} \) as its adjacent sides.

• The direction of the cross product is determined by the right-hand rule.
• The cross product is only defined in three dimensions.
• Unlike the dot product, the cross product of two vectors is a vector, not a scalar.

The cross product is not commutative, meaning \( \mathbf{a} \times \mathbf{b} eq \mathbf{b} \times \mathbf{a} \), instead \( \mathbf{b} \times \mathbf{a} = - (\mathbf{a} \times \mathbf{b}) \).
The vector triple product identity is pivotal for solving complex vector problems. It transforms the cross product of a cross product into a combination of dot products and vector additions.Utilizing this identity, \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) shows how cross products interact with dot products.

Exploring Vector Spaces

Vector spaces lay the groundwork for understanding operations like the cross product and the dot product.A vector space is a collection of vectors that can be added together and multiplied by scalars to produce another vector within the same space.
Vectors themselves are directed line segments, often used in physics to represent quantities like force and velocity.

Some key properties of vector spaces include:

• Closure under addition and scalar multiplication.
• There exists a zero vector that serves as an additive identity.
• Vectors have additive inverses (negative vectors).
• Associativity and commutativity of addition.
• Distributive properties for vector and scalar multiplication.

These properties show how vectors interact and the rules they obey within the space.When dealing with a plane defined by two vectors—such as \( \mathbf{a} \) and \( \mathbf{b} \)—any linear combination, like \( c_1 \mathbf{a} + c_2 \mathbf{b} \), lies in the same plane.
This characteristic is used to demonstrate how the vector triple product \((\mathbf{a} \times \mathbf{b}) \times \mathbf{c}\) remains within the plane formed by \( \mathbf{a} \) and \( \mathbf{b} \), assuming \( \mathbf{a} \) and \( \mathbf{b} \) are non-parallel.

The Dot Product's Role

The dot product, also known as the scalar product, is another essential operation in vector algebra.It involves two vectors and results in a scalar quantity, as opposed to a vector output like in the cross product.
The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{c} \) is calculated as \( \mathbf{a} \cdot \mathbf{c} = |\mathbf{a}| |\mathbf{c}| \cos \theta \), where \( \theta \) is the angle between the vectors.

Some important aspects of the dot product include:

• It is commutative, meaning \( \mathbf{a} \cdot \mathbf{c} = \mathbf{c} \cdot \mathbf{a} \).
• Produces a measure of how much one vector goes in the direction of another.
• Is zero if the vectors are perpendicular, indicating no directional similarity.

In the context of the vector triple product identity, the dot product plays a crucial role.It helps in breaking down the complex vector interaction into simpler components that are linear combinations of the base vectors. This is seen where the final expression \((\mathbf{a} \cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a}\) uses dot products to scale \( \mathbf{b} \) and \( \mathbf{a} \).
This ultimately demonstrates how the result remains within the plane defined by those vectors, providing clarity to vector relationships.