Problem 50 - Advanced Engineering Mathematics

Question

Use the dot product to provethe Cauchy-Schwarz inequality: \(|\mathbf{a} \cdot \mathbf{b}| \leq\|\mathbf{a}\|\|\mathbf{b}\|\).

Step-by-Step Solution

Answer

The Cauchy-Schwarz inequality \(|\mathbf{a} \cdot \mathbf{b}| \leq \|\mathbf{a}\|\|\mathbf{b}\|\) is proved using the definition of dot product and an angle-based interpretation.

Step 1: Define the Dot Product and Magnitude

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ ... + a_nb_n \). The magnitude of a vector \( \mathbf{a} \) is \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + \ ... + a_n^2} \). Similarly, the magnitude of \( \mathbf{b} \) is \( \|\mathbf{b}\| = \sqrt{b_1^2 + b_2^2 + \ ... + b_n^2} \).

Step 2: Express Vector Dot Product Using Angle

The dot product can also be expressed in terms of the cosine of the angle \( \theta \) between the two vectors: \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta \).

Step 3: Interpret Using Trigonometry

Since \( \cos \theta \) satisfies \(-1 \leq \cos \theta \leq 1\), it follows that \( -\|\mathbf{a}\|\|\mathbf{b}\| \leq \mathbf{a} \cdot \mathbf{b} \leq \|\mathbf{a}\| \|\mathbf{b}\| \).

Step 4: Establish the Cauchy-Schwarz Inequality

Taking absolute values gives \( |\mathbf{a} \cdot \mathbf{b}| = |\|\mathbf{a}\| \|\mathbf{b}\| \cos \theta| \), hence \( |\mathbf{a} \cdot \mathbf{b}| \leq \|\mathbf{a}\| \|\mathbf{b}\| \).

Step 5: Conclusion

Thus, by showing that the absolute value of the dot product does not exceed the product of the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \), we've proved the Cauchy-Schwarz inequality: \( |\mathbf{a} \cdot \mathbf{b}| \leq \|\mathbf{a}\| \|\mathbf{b}\| \).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It involves two vectors and results in a scalar, or single number. For two vectors \( \mathbf{a} = (a_1, a_2, ..., a_n) \) and \( \mathbf{b} = (b_1, b_2, ..., b_n) \), the dot product is calculated as:

\( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \ ... + a_nb_n \)

This operation tells us how much of one vector goes in the direction of another. It has practical applications in physics and engineering, representing quantities like work and projections.
A significant property of the dot product is its relation to the angle between vectors, which is a key concept when using it to derive inequalities like the Cauchy-Schwarz Inequality.

Vector Magnitude

The magnitude of a vector, often referred to as its length or norm, quantifies the size of the vector. For a vector \( \mathbf{a} = (a_1, a_2, ..., a_n) \), its magnitude is calculated using:

\( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2 + \ ... + a_n^2} \)

This formula is a straightforward extension of the Pythagorean theorem into higher dimensions.
The magnitude is a non-negative value that provides essential geometric information about the vector. Understanding how to compute vector magnitudes is crucial in vector geometry because it helps in comparing vectors and understanding their interactions in spaces, as seen in the Cauchy-Schwarz Inequality.

Trigonometric Functions

In the context of vectors, trigonometric functions, particularly cosine, play a crucial role in relating the dot product to the angle between vectors. The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed as:

\( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta \)

Here, \( \theta \) is the angle between the vectors. The cosine function, which varies between -1 and 1, dictates how the vectors align with each other.

• If \( \cos \theta = 1 \), the vectors are completely aligned, pointing in the same direction.
• If \( \cos \theta = -1 \), they are opposite.
• If \( \cos \theta = 0 \), they are perpendicular.

This relationship is fundamental in understanding vector interactions and is a cornerstone in proving vector inequalities like Cauchy-Schwarz.

Vector Geometry

Vector geometry is the study of geometrical structures using vectors. It provides a robust framework for understanding spatial relationships and orientations in both two-dimensional and three-dimensional spaces. Vectors can represent lines, surfaces, directions, and quantities with both magnitude and direction.
In vector geometry, important concepts like the dot product and vector magnitude help in understanding distances, angles, and projections. These concepts are used to explore properties like orthogonality, parallelism, and even inequalities, such as the Cauchy-Schwarz Inequality. By analyzing vectors geometrically, one can gain insights into more complex structures and their spatial behavior.
An essential part of vector geometry is visualizing how different vectors interact, which helps in solving real-world problems in physics, computer graphics, and more.