Question

Another operation with vectors is the scalar triple product, defined to be \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}),\) for vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) in \(\mathbb{R}^{3}\) Prove that \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)

Short answer

Question: Prove that for vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) in \(\mathbb{R}^{3}\), the scalar triple product \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w})\) is equal to \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\). Answer: To prove this, expand both expressions using vector components, then compare the two expressions and show that they are equal. Upon inspection, you will find that the terms are rearranged but equal in both expressions. Thus, it can be concluded that \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\).

Step-by-step solution

Write down the given expression

The given expression is: \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)

Expand the expressions using vector components

Let \(\mathbf{u}=(u_1, u_2, u_3)\), \(\mathbf{v}=(v_1, v_2, v_3)\), and \(\mathbf{w}=(w_1, w_2, w_3)\). We can expand the expressions as follows: - \(\mathbf{v} \times \mathbf{w} = (v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1)\) - \(\mathbf{u} \times \mathbf{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)\) Now, we can find the dot products: - \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) = u_1(v_2w_3 - v_3w_2) + u_2(v_3w_1 - v_1w_3) + u_3(v_1w_2 - v_2w_1)\) - \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = (u_2v_3 - u_3v_2)w_1 + (u_3v_1 - u_1v_3)w_2 + (u_1v_2 - u_2v_1)w_3\)

Show that the two expressions are equal

Let's compare the two expressions to show that they are equal: \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) = u_1(v_2w_3 - v_3w_2) + u_2(v_3w_1 - v_1w_3) + u_3(v_1w_2 - v_2w_1)\) \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = (u_2v_3 - u_3v_2)w_1 + (u_3v_1 - u_1v_3)w_2 + (u_1v_2 - u_2v_1)w_3\) Upon inspection, we can see that the terms are rearranged but equal: - \(u_1(v_2w_3 - v_3w_2) = (u_2v_3 - u_3v_2)w_1\) - \(u_2(v_3w_1 - v_1w_3) = (u_3v_1 - u_1v_3)w_2\) - \(u_3(v_1w_2 - v_2w_1) = (u_1v_2 - u_2v_1)w_3\) Thus, the two expressions are equal, and we have shown that \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\).

Key concepts

Vector Operations in Calculus

In calculus, understanding vector operations is crucial for analyzing directions and magnitudes in multi-dimensional spaces. One of the fundamental operations is the scalar triple product, which involves a combination of the dot product and the cross product. It can be visually interpreted as the volume of the parallelepiped formed by three vectors. To fully grasp such concepts, it's essential to be comfortable with basic operations like vector addition, scalar multiplication, and the aforementioned dot and cross products. These operations allow us to solve problems in physics and engineering, such as determining the work done by a force or the torque on a rotating system.

When calculating the scalar triple product, we examine the interplay between vectors in three-dimensional space, \( \mathbb{R}^3 \). This process involves projecting one vector onto the perpendicular plane formed by the other two vectors and provides insights into the orthogonality and linear dependence of the vectors involved.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is denoted by a dot \( \cdot \) and is calculated as the sum of the products of the corresponding entries of the two sequences of numbers. The dot product formula for two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \), where \( a_i \) and \( b_i \) are components of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

The dot product has geometric significance as well; it can be used to determine the angle between two vectors and whether they are orthogonal (perpendicular to each other). If the dot product of two vectors is zero, they are considered orthogonal.

Cross Product

The cross product, also known as the vector product, contrasts with the dot product by returning a vector instead of a scalar. This operation is applied to two vectors in three-dimensional space and results in a vector that is perpendicular to the plane which contains the original vectors. Its length corresponds to the area of the parallelogram that the vectors span.

The cross product of vectors \( \mathbf{a} \) and \( \mathbf{b} \) is denoted by \( \mathbf{a} \times \mathbf{b} \) and is calculated using the components of the vectors: \( \mathbf{a} \times \mathbf{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \). This vector operation is essential in physics for calculating rotational forces and magnetic fields, exemplifying the profound relationship between vector mathematics and physical phenomena.

Vector Components

Vectors in a three-dimensional space are described by their components along the x, y, and z axes. These components are the building blocks for all vector operations and serve as a bridge between geometric interpretation and algebraic calculation.

A vector \( \mathbf{v} \) in three-dimensional space can be represented as \( \mathbf{v} = (v_1, v_2, v_3) \) where \( v_1, v_2, \) and \( v_3 \) are the magnitudes of the vector in the direction of the x, y, and z axes respectively. Vector components allow us to perform calculations such as addition, subtraction, and multiplication (dot and cross product) in a straightforward component-wise manner. Moreover, they give us a way to visualize vectors in a coordinate system, aiding in understanding concepts such as vector projection, vector fields, and even more complex operations like the scalar triple product.