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 \(\mathbb{R}^{3}\). $$\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})$$

Short answer

Answer: Yes, the given statement is true for all \(\mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w}\) in \(\mathbb{R}^{3}\).

Step-by-step solution

Examine the given expressions

We are given the following two expressions: i. \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})\) ii. \(\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})\) Our goal is to prove or disprove that these two expressions are equal.

Apply the vector triple product rule

Recall that one of the rules for vector triple product states that for any three vectors \(\mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c}\): $$ (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a}\cdot \mathbf{c})\mathbf{b} - (\mathbf{b} \cdot \mathbf{c})\mathbf{a} $$ Using this rule, we can expand the expression \(\mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})\): \((\mathbf{u} \times \mathbf{v}) \times \mathbf{w} = (\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u}\) Now take the dot product with \(\mathbf{w}\): \(\mathbf{w} \cdot((\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u})\)

Apply the properties of dot products

Using the properties of dot products (commutativity and distribution), we can rewrite the expression from Step 2: \(\mathbf{w} \cdot((\mathbf{u} \cdot \mathbf{w})\mathbf{v} - (\mathbf{v} \cdot \mathbf{w})\mathbf{u}) = (\mathbf{u} \cdot \mathbf{w})(\mathbf{v} \cdot \mathbf{w}) - (\mathbf{v} \cdot \mathbf{w})(\mathbf{u} \cdot \mathbf{w})\) Now, we can see that this expression is equal to the first expression: \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) = \mathbf{w} \cdot(\mathbf{u} \times \mathbf{v})\)

Conclusion

We have proved that the given statement is true for all \(\mathbf{u}, \mathbf{v}, \text{ and } \mathbf{w}\) in \(\mathbb{R}^{3}\).

Key concepts

Vector Triple Product

The vector triple product is a fascinating concept in vector calculus that involves taking the cross product of three vectors in sequence. Specifically, it describes the cross product of the cross product of two vectors, then crossed with a third vector. In mathematical terms, for vectors \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) in \( \mathbb{R}^3 \), it is defined as:

• \( (\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{b} \cdot \mathbf{c}) \mathbf{a} \)

This result is known as the vector triple product expansion or the BAC-CAB rule. It simplifies the process of evaluating such expressions and reveals the underlying geometric relationships among vectors. The importance of this rule cannot be overstated as it frequently appears in physics and engineering problems, including those involving torque and magnetic forces. Understanding this rule allows us to transform complex vector expressions into more manageable and insight-bearing forms.

Dot Product

The dot product, also known as the scalar product, is an operation that combines two vectors into a single scalar quantity. For any vectors \( \mathbf{a} \) and \( \mathbf{b} \) in \( \mathbb{R}^3 \), it is denoted as \( \mathbf{a} \cdot \mathbf{b} \) and calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \) when expressed component-wise.
• Alternatively, it can be calculated as \( \|\mathbf{a}\| \|\mathbf{b}\| \cos\theta \), where \( \theta \) is the angle between the two vectors.

The dot product is particularly useful because it provides information about the angle between vectors and their relative orientation. If the dot product is zero, the vectors are orthogonal, emphasizing the perpendicular nature in geometric contexts. It's a central tool in projecting vectors onto one another, finding work done by a force, and determining the component of one vector in the direction of another.

Cross Product

The cross product is another fundamental vector operation in \( \mathbb{R}^3 \). Unlike the dot product, the cross product of two vectors results in another vector, which is perpendicular to both of the original vectors. Given vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the cross product \( \mathbf{a} \times \mathbf{b} \) is calculated as:

• \( \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \)

The magnitude of the cross product gives the area of the parallelogram formed by the two vectors, signifying how the interaction of these vectors extends in space. This operation is widely used in physics to determine quantities such as torque and angular momentum. The direction of the resulting vector can be determined using the right-hand rule, adding a helpful geometric understanding to the cross product's utility.

Vectors in \(\mathbb{R}^{3}\)

Vectors in \( \mathbb{R}^3 \) are essential objects in mathematics and science describing quantities with both magnitude and direction. These vectors are typically expressed as \( \mathbf{v} = (v_1, v_2, v_3) \), with each component corresponding to a spatial dimension—commonly the x, y, and z axes.

• Magnitude: The magnitude or length of a vector \( \mathbf{v} \) is given by \( \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \).
• Direction: The direction is encoded by its components, which can be converted into unit vectors by normalizing the vector.
• Operations: Fundamental operations between vectors include addition, subtraction, scalar multiplication, dot products, and cross products.

Vectors are used to represent physical quantities like velocity, force, and acceleration, offering a robust way to model real-world scenarios. The familiarity with their manipulation and understanding their geometric interpretations is crucial for solving a broad range of problems in engineering, physics, and computer graphics.