Problem 22 - Essential Calculus Early Transcendentals

Question

Show that \((\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b}=0\) for all vectors a and \(\mathbf{b}\) in \(V_{3}\).

Step-by-Step Solution

Answer

The cross product \\(\\mathbf{a} \\times \\mathbf{b}\\) is perpendicular to \\mathbf{b}\\, making their dot product zero.

Step 1: Understanding Cross and Dot Product Basics

The expression \(\mathbf{a} \times \mathbf{b}\) denotes the cross product of two vectors \mathbf{a}\ and \mathbf{b}\. The cross product results in a vector that is perpendicular to both \mathbf{a}\ and \mathbf{b}\. The dot product of two vectors results in a scalar (a single number). To show that \(\mathbf{a} \times \mathbf{b}\) \cdot \mathbf{b} = 0\, we must demonstrate that the vector from the cross product is perpendicular to \mathbf{b}\.

Step 2: Applying the Perpendicularity Concept

For two vectors \mathbf{c}\ and \mathbf{d}\, if \mathbf{c} \cdot \mathbf{d} = 0\, then \mathbf{c}\ is perpendicular to \mathbf{d}\. By the definition of the cross product, \(\mathbf{a} \times \mathbf{b}\)\ is perpendicular to both \mathbf{a}\ and \mathbf{b}\. Therefore, the dot product of \mathbf{b}\ with any vector perpendicular to it, such as \(\mathbf{a} \times \mathbf{b}\)\, must be zero.

Step 3: Validation through Mathematical Definition

The mathematical definition of the cross product states that the magnitude of \(\mathbf{a} \times \mathbf{b}\)\ is \left|\mathbf{a}\right| \left|\mathbf{b}\right|\sin(\theta)\, where \(\theta\) is the angle between \mathbf{a}\ and \mathbf{b}\. The direction of this vector is perpendicular to both. Since the dot product of a vector with another vector perpendicular to it is always zero, \(\mathbf{a} \times \mathbf{b}\) \cdot \mathbf{b} = 0\.

Key Concepts

Cross Product

The cross product is an essential operation involving two vectors in three-dimensional space. When you take the cross product of vectors \(\mathbf{a}\) and \(\mathbf{b}\), the result is another vector. The unique characteristic of this new vector is that it is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\). To compute the cross product \(\mathbf{a} \times \mathbf{b}\), you can use the determinant of a matrix that involves the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), and the components of \(\mathbf{a}\) and \(\mathbf{b}\).

• The formula for the cross product is:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]
• This expands to:\[(a_2 b_3 - a_3 b_2) \mathbf{i} - (a_1 b_3 - a_3 b_1) \mathbf{j} + (a_1 b_2 - a_2 b_1) \mathbf{k}\]

The significance of the cross product comes from its perpendicular nature, particularly in physical applications like torque and angular momentum where direction is crucial.

Dot Product

The dot product is a basic and widely used operation on two vectors yielding a scalar. If you have vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product \(\mathbf{a} \cdot \mathbf{b}\) communicates the extent to which two vectors are aligned. The resulting scalar gives a measure of this alignment.

• The formula for the dot product is:\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\]
• An alternative expression relates it to the cosine of the angle \(\theta\) between the vectors:\[\mathbf{a} \cdot \mathbf{b} = \left|\mathbf{a}\right| \left|\mathbf{b}\right| \cos(\theta)\]

In essence, the dot product shows how much of one vector reaches in the direction of another. It helps recognize perpendicular vectors as for them, the dot product will be zero, indicating no alignment.

Perpendicular Vectors

Perpendicular vectors occur frequently in vector calculus and indicate a special relationship between two vectors: they meet at a right angle. When vectors \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular (orthogonal), their dot product is zero: \(\mathbf{a} \cdot \mathbf{b} = 0\) because the cosine of a 90-degree angle is zero.

• This characteristic has practical implications:
  • In geometry, it defines orthogonal systems.
  • In physics, it underpins analyses where forces or movements are independent.

The cross product of two vectors involves a perpendicular vector to the original pair, reflecting their orthogonality. Thus, when calculating \( (\mathbf{a} \times \mathbf{b}) \cdot \mathbf{b} = 0 \), it utilizes this inherent orthogonality property. Understanding the relationship between perpendicular vectors strengthens comprehension of vector interactions and operations.