Question

Determine whether the following statements are true using a proof or counterexample. Assume that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are nonzero vectors in \(\mathbb{R}^{3}\). $$\mathbf{u} \times(\mathbf{u} \times \mathbf{v})=\mathbf{0}$$

Short answer

Answer: No, the expression \(\mathbf{u} \times(\mathbf{u} \times \mathbf{v})\) is not equal to the zero vector, as shown by the provided analysis and counterexample.

Step-by-step solution

Recall the properties of cross product

Recall that the cross product of two vectors results in a vector that is orthogonal (perpendicular) to both of the original vectors.

Expand the given expression

We are given the expression \(\mathbf{u} \times(\mathbf{u} \times \mathbf{v})\). We know that \((\mathbf{u} \times \mathbf{v})\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). Let's denote this orthogonal vector as \(\mathbf{w}\). Then the expression becomes \(\mathbf{u} \times \mathbf{w}\).

Determine if the resulting cross product is zero

Now we need to determine if the cross product \(\mathbf{u} \times \mathbf{w}\) is equal to the zero vector. Recall that \(\mathbf{w}\) is orthogonal to \(\mathbf{u}\) by definition. The cross product of two orthogonal vectors results in a zero vector because the sine of the angle between the orthogonal vectors will be equal to 1 (since the angle between them is 90 degress or \(\pi/2\) radians) and the magnitude of the cross product is equal to the product of the magnitudes multiplied by the sine of the angle between them: $$|\mathbf{u} \times \mathbf{w}| = |\mathbf{u}| \cdot |\mathbf{w}| \cdot \sin(\pi/2)$$ Since \(\sin(\pi/2) = 1\), the resulting magnitude will be nonzero (because both \(\mathbf{u}\) and \(\mathbf{w}\) are nonzero vectors). This means that the cross product \(\mathbf{u} \times \mathbf{w}\) is not equal to the zero vector, and our statement is not true.

Provide a counterexample

Let's provide a counterexample to prove that the given statement is not true. Suppose we have the following nonzero vectors: $$\mathbf{u} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix},\ \mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$$ First, find the cross product of \(\mathbf{u}\) and \(\mathbf{v}\): $$\mathbf{w} = \mathbf{u} \times \mathbf{v} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$$ Now, find the cross product of \(\mathbf{u}\) and \(\mathbf{w}\): $$\mathbf{u} \times \mathbf{w} = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \times \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ -1 \\ 0 \end{pmatrix}$$ Since \(\mathbf{u} \times(\mathbf{u} \times \mathbf{v}) \neq \mathbf{0}\), the given statement is indeed not true.

Key concepts

Vector Multiplication

Vector multiplication, specifically the cross product, is a mathematical operation that involves two vectors in three-dimensional space. To calculate the cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \), denoted as \( \mathbf{a} \times \mathbf{b} \), the result is a new vector that is orthogonal to both \( \mathbf{a} \) and \( \mathbf{b}\).

The magnitude of this new vector is given by \( |\mathbf{a} \times \mathbf{b}| = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \sin(\theta) \) where \( \theta \) is the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). The direction of the resulting vector is determined by the right-hand rule, which states that if you point your right thumb in the direction of \( \mathbf{a} \) and your fingers in the direction of \( \mathbf{b}\), your palm will face the direction of \( \mathbf{a} \times \mathbf{b}\).

Importance in Mathematics and Physics

Cross product is pivotal in understanding rotational vectors and torque in physics, as well as in determining the area of parallelograms formed by two vectors. However, the result of a cross product is not commutative; switching the order of multiplication yields the opposite direction of the resulting vector, making it critical to maintain the correct sequence of vectors being multiplied.

Orthogonal Vectors

Orthogonal vectors are two vectors that meet at a right angle, meaning the angle between them is 90 degrees or \( \pi/2 \) radians. In terms of the dot product, two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are orthogonal if their dot product is zero, \( \mathbf{a} \cdot \mathbf{b} = 0 \).

When vectors are orthogonal, their cross product magnitude reaches its maximum value because \( \sin(\theta) \) is at its peak when \( \theta = \pi/2 \). This concept is crucial for understanding the geometry of three-dimensional space and is often used in computer graphics, engineering, and physics to determine perpendicularity and stability.

Orthogonality in Vector Spaces

Beyond three-dimensional space, the concept of orthogonality extends to more abstract vector spaces in linear algebra. In these higher-dimensional spaces, 'orthogonal' still refers to vectors that have no component in the same direction, a fundamental idea necessary for topics such as eigenvalues and eigenvectors, as well as many optimization problems. Identifying orthogonal vectors can simplify computations and resolve complex geometrical arrangements.

Counterexample in Mathematics

A counterexample in mathematics is a specific case for which a general statement is shown to be false. This method is employed particularly in the field of abstract reasoning, where the truth value of statements is often established by the absence of any counterexample.

For instance, when presented with a statement that applies to all elements of a set, finding just one element for which the statement fails proves that the statement cannot be universally true. In the context of vector multiplication, providing a real-world vector pair that when applied to the given claim results in a contradiction serves as a potent demonstration. Counterexamples are often more straightforward and concise than formal proofs of negation, making them a powerful tool for disproving conjectures and theorems.

Role in Mathematical Discourse

The search for counterexamples fosters critical thinking and a deeper understanding of the underlying principles. It is also a means to test the boundaries and limitations of mathematical theories. In education, counterexamples are used to challenge students and allow them to explore the breadth and depth of mathematical concepts, solidifying their grasp through the analysis of why a particular case breaks the rule.