Problem 31 - Thomas Calculus

Question

Let \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) be vectors. Which of the following make sense, and which do not? Give reasons for your answers. $$ \begin{array}{ll}{\text { a. }(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}} & {\text { b. } \mathbf{u} \times(\mathbf{v} \cdot \mathbf{w})} \\\ {\mathbf{c} . \mathbf{u} \times(\mathbf{v} \times \mathbf{w})} & {\text { d. } \mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})}\end{array} $$

Step-by-Step Solution

Answer

a and c are valid; b and d are not valid.

Step 1: Understanding Cross and Dot Products

The cross product \( \mathbf{u} \times \mathbf{v} \) results in a vector that is perpendicular to both \( \mathbf{u} \) and \( \mathbf{v} \), and is only defined for three-dimensional vectors. The dot product \( \mathbf{u} \cdot \mathbf{v} \) results in a scalar and is defined for vectors of the same dimension.

Step 2: Evaluate Expression a: \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\)

\((\mathbf{u} \times \mathbf{v}) \) is a vector. Since \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} )\) is a dot product between two vectors, it is a valid expression, resulting in a scalar.

Step 3: Evaluate Expression b: \(\mathbf{u} \times (\mathbf{v} \cdot \mathbf{w})\)

\((\mathbf{v} \cdot \mathbf{w})\) is a scalar. Since we cannot take a cross product between a vector and a scalar, this expression is not valid.

Step 4: Evaluate Expression c: \((\mathbf{u} \times (\mathbf{v} \times \mathbf{w})\)

\((\mathbf{v} \times \mathbf{w})\) is a vector, so \(\mathbf{u} \times (\mathbf{v} \times \mathbf{w}) \) is defined as the cross product of two vectors, resulting in another vector. Therefore, this expression is valid.

Step 5: Evaluate Expression d: \(\mathbf{u} \cdot (\mathbf{v} \cdot \mathbf{w})\)

\(\mathbf{v} \cdot \mathbf{w}\) results in a scalar. However, the dot product \(\mathbf{u} \cdot (\text{scalar})\) is not defined because the dot product should be between two vectors. Thus, this expression is not valid.

Key Concepts

Cross Product

The cross product is an essential operation in vector mathematics, particularly for three-dimensional vectors. Calculating the cross product of two vectors results in another vector that is orthogonal, or perpendicular, to the original two. This means if you have vectors \( \mathbf{u} \) and \( \mathbf{v} \), the cross product \( \mathbf{u} \times \mathbf{v} \) will give you a vector pointing out of the plane formed by \( \mathbf{u} \) and \( \mathbf{v} \).
The formula used to find the cross product is based on a determinant and can be expressed as:

• \( \mathbf{u} = \langle a_1, b_1, c_1 \rangle \)
• \( \mathbf{v} = \langle a_2, b_2, c_2 \rangle \)
• \( \mathbf{u} \times \mathbf{v} = \langle b_1c_2 - c_1b_2, c_1a_2 - a_1c_2, a_1b_2 - b_1a_2 \rangle \)

However, it's essential to note that the cross product is specifically defined for vectors in three-dimensional space. Attempting to compute a cross product involving a scalar (a single number) does not make sense, as scalars lack direction, which is crucial for determining perpendicularity.

Dot Product

The dot product is another fundamental operation in vector calculations, defined for two vectors of the same dimension, producing a scalar. Unlike the cross product, the dot product measures how much one vector goes in the direction of another vector, hence often being described as a measure of similarity in direction.
The dot product of vectors \( \mathbf{u} = \langle a_1, b_1, c_1 \rangle \) and \( \mathbf{v} = \langle a_2, b_2, c_2 \rangle \) can be calculated as follows:

• \( \mathbf{u} \cdot \mathbf{v} = a_1a_2 + b_1b_2 + c_1c_2 \)

This operation results in a scalar value, meaning it's a single number rather than another vector. A good way to visualize this operation is considering if the result is zero, which indicates that the vectors are orthogonal (perpendicular) to one another. The dot product cannot be computed between a scalar and a vector, which is why expressions expecting a dot product input from a scalar are considered invalid.

Vectors

Vectors are fundamental building blocks used in physics and mathematics to represent quantities with both magnitude and direction. Unlike scalars that only have magnitude (like mass or temperature), vectors encompass a directional component, making them ideal for describing forces, velocities, and more in space.
Each vector is typically represented in Cartesian (rectangular) coordinates, such as \( \mathbf{v} = \langle a, b, c \rangle \), where \( a, b, \) and \( c \) are components of the vector along the x, y, and z axes, respectively. This setup is straightforward for calculations and visualizations in three dimensions.
Vectors are not limited to describing physical motion; they also serve as fundamental components in various mathematical operations. Operations on vectors like addition, subtraction, scalar multiplication, dot product, and cross product are manipulations that help solve complex problems in engineering, computer graphics, and more.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar value and is a straightforward operation that scales the vector's magnitude without altering its direction. If you have a vector \( \mathbf{v} = \langle a, b, c \rangle \) and a scalar \( k \), the result of this operation is:

• \( k\mathbf{v} = \langle ka, kb, kc \rangle \)

This process stretches or compresses the vector, depending on the magnitude of the scalar. If the scalar is positive, the vector points in the same direction, but if it is negative, the vector's direction is reversed.
Scalar multiplication is crucial in many applications, allowing for simple yet powerful transformations of vectors in both mathematics and physical sciences. This operation maintains the vector's direction but alters its length, proving invaluable in adjusting vector-based representations within applied mathematics and computational fields.