Problem 43 - APEX Calculus

Question

Show, using the definition of the Cross Product, that \(\vec{u} \cdot(\vec{u} \times\) \(\vec{v})=0 ;\) that is, that \(\vec{u}\) is orthogonal to the cross product of \(\vec{u}\) and \(\vec{v} .\)

Step-by-Step Solution

Answer

The vector \( \vec{u} \) is orthogonal to \( \vec{u} \times \vec{v} \), so the dot product is 0.

Step 1: Understanding the Problem

We need to show that the dot product \( \vec{u} \cdot (\vec{u} \times \vec{v}) = 0 \). This means \( \vec{u} \) is orthogonal to its own cross product with another vector \( \vec{v} \).

Step 2: Definition of Cross Product

The cross product \( \vec{u} \times \vec{v} \) results in a vector that is perpendicular to both \( \vec{u} \) and \( \vec{v} \) as per definition. This implies that any vector dotted with its perpendicular vector will result in a scalar of 0.

Step 3: Understanding Dot Product of Orthogonal Vectors

The dot product \( \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \) for vectors \( \vec{a} = (a_x, a_y, a_z) \) and \( \vec{b} = (b_x, b_y, b_z) \). If the vectors are orthogonal (perpendicular), then \( \vec{a} \cdot \vec{b} = 0 \).

Step 4: Apply to Given Vectors

Since \( \vec{u} \times \vec{v} \) is perpendicular to \( \vec{u} \), we can directly apply the property of dot products with orthogonal vectors. Thus,\[ \vec{u} \cdot (\vec{u} \times \vec{v}) = 0 \].

Key Concepts

Dot Product in Vector Algebra

The dot product, also known as the scalar product, is an essential operation in vector algebra. It involves two vectors and results in a single scalar value. If you have two vectors \( \vec{a} = (a_x, a_y, a_z) \) and \( \vec{b} = (b_x, b_y, b_z) \), their dot product is calculated as \( \vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \). This operation measures how much one vector extends in the direction of another.
Some key properties of the dot product include:

• Commutative: \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \)
• Distributive: \( \vec{a} \cdot (\vec{b} + \vec{c}) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} \)
• Produces zero if vectors are orthogonal

This last property is crucial because it hints at the relationship between vectors' orientations. If two vectors are perpendicular, their dot product will always be zero, aiding in our understanding of orthogonality.

Understanding Orthogonal Vectors

Vectors are said to be orthogonal when they are perpendicular to each other. This is a special case in vector algebra where the angle between the vectors is 90 degrees. Perpendicular vectors have the interesting property that their dot product is always zero. Orthogonal vectors, therefore, showcase an absence of influence in each other's direction.
Understanding this concept is particularly useful when dealing with vector components in different dimensions. Whenever you compute the dot product of two vectors and the result is zero, this indicates they are orthogonal.
In the context of the original exercise, the vector \( \vec{u} \) is shown to be orthogonal to its own cross product with vector \( \vec{v} \), because the dot product \( \vec{u} \cdot (\vec{u} \times \vec{v}) = 0 \). This demonstrates the power and application of recognizing orthogonal vectors through dot products.

Exploring Vector Algebra

Vector algebra is a branch of mathematics dealing with quantities that have both magnitude and direction. It encompasses various operations like the addition, subtraction, and multiplication of vectors through dot and cross products. These operations offer profound insights into geometric interpretations.
For example, the cross product—not to be confused with the dot product—results in another vector that is perpendicular to the original two vectors. It is especially useful in determining the area of parallelograms and in physics for torque calculations.
By mastering these operations, you can solve complex problems involving spatial relationships and vector fields. These concepts form the backbone of three-dimensional modeling, physics, engineering, and computer graphics, among other fields. Understanding both the dot and cross product operations significantly enhances your ability to manipulate and interpret vector data in scientific and practical applications.