Question

Cauchy-Schwarz Inequality The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\) ). This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Triangle Inequality Consider the vectors \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{u}+\mathbf{v}\) (in any number of dimensions). Use the following steps to prove that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) a. Show that \(|\mathbf{u}+\mathbf{v}|^{2}=(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}+\mathbf{v})=|\mathbf{u}|^{2}+\) \(2 \mathbf{u} \cdot \mathbf{v}+|\mathbf{v}|^{2}\) b. Use the Cauchy-Schwarz Inequality to show that \(|\mathbf{u}+\mathbf{v}|^{2} \leq(|\mathbf{u}|+|\mathbf{v}|)^{2}\) c. Conclude that \(|\mathbf{u}+\mathbf{v}| \leq|\mathbf{u}|+|\mathbf{v}|\) d. Interpret the Triangle Inequality geometrically in \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\).

Short answer

Based on the step-by-step solution provided, write a short answer discussing the proof of the Triangle Inequality using the Cauchy-Schwarz Inequality. The Triangle Inequality, defined as \(|\mathbf{u}+\mathbf{v}| \leq |\mathbf{u}|+|\mathbf{v}|\), is proven using the Cauchy-Schwarz Inequality, \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\). The proof is as follows: 1. Expand the magnitude squared of the vector sum, obtaining \(|\mathbf{u}+\mathbf{v}|^{2} = |\mathbf{u}|^{2} + 2\mathbf{u}\cdot\mathbf{v} + |\mathbf{v}|^{2}\). 2. Apply the Cauchy-Schwarz Inequality, resulting in \(|\mathbf{u}+\mathbf{v}|^{2} \leq (|\mathbf{u}| + |\mathbf{v}|)^{2}\). 3. Take the square root of both sides to get the Triangle Inequality, \(|\mathbf{u}+\mathbf{v}| \leq |\mathbf{u}|+|\mathbf{v}|\). Geometrically, this means that the length of any side of a triangle in 2D or 3D space is less than or equal to the sum of the lengths of the other two sides. The proof demonstrates the fundamental property of triangles in Euclidean space, ensuring that the side represented by the vector sum \(\mathbf{u}+\mathbf{v}\) has a length less than or equal to the sum of the lengths of the other two sides, \(\mathbf{u}\) and \(\mathbf{v}\).

Step-by-step solution

Step A: Expand Vector Magnitude Squared

To prove the Triangle Inequality, we first need to find the magnitude squared of the vector sum \(\mathbf{u}+\mathbf{v}\). Using the dot product property, we have: \(|\mathbf{u}+\mathbf{v}|^{2} = (\mathbf{u}+\mathbf{v})\cdot(\mathbf{u}+\mathbf{v})\) Expanding the dot product, we get: \(|\mathbf{u}+\mathbf{v}|^{2} = |\mathbf{u}|^{2} + 2\mathbf{u}\cdot\mathbf{v} + |\mathbf{v}|^{2}\)

Step B: Apply Cauchy-Schwarz Inequality

Now, we will use the Cauchy-Schwarz Inequality, which states that \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\): \(\phantom{=} |\mathbf{u}|^{2} + 2\mathbf{u}\cdot\mathbf{v} + |\mathbf{v}|^{2} \leq |\mathbf{u}|^{2} + 2|\mathbf{u}||\mathbf{v}| + |\mathbf{v}|^{2}\) The inequality above is equivalent to: \(|\mathbf{u}+\mathbf{v}|^{2} \leq (|\mathbf{u}| + |\mathbf{v}|)^{2}\)

Step C: Conclude Triangle Inequality

Since both sides of the inequality are non-negative, we can safely take the square root to obtain the Triangle Inequality: \(|\mathbf{u}+\mathbf{v}| \leq |\mathbf{u}|+|\mathbf{v}|\)

Step D: Geometrical Interpretation of Triangle Inequality

In the context of \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3}\) (2D or 3D spaces), the Triangle Inequality states that the length of any one side of a triangle is less than or equal to the sum of the lengths of the other two sides. In other words, if we consider \(\mathbf{u}\) and \(\mathbf{v}\) as two sides of a triangle, their vector sum \(\mathbf{u}+\mathbf{v}\) represents the third side. The Triangle Inequality ensures that this side has a length less than or equal to the sum of the lengths of the other two sides, which is a fundamental property of triangles in Euclidean space.

Key concepts

Vector Magnitude

Vector magnitude is a fundamental concept in the study of vectors, representing the length of the vector in space. Every vector in a Euclidean space has a magnitude, which can be calculated using the formula:
\[ |\mathbf{v}| = \sqrt{v_1^2+v_2^2+...+v_n^2} \]
where \(v_1, v_2, ..., v_n\) are the components of the vector \(\mathbf{v}\). Magnitude is always non-negative and provides information about the size of the vector regardless of its direction. Understanding vector magnitude is crucial for grasping concepts like the Cauchy-Schwarz Inequality and Triangle Inequality, where the length of vectors is compared or combined.

Dot Product Property

The dot product, also known as the scalar product, is an operation that takes two vectors and returns a single number (scalar). The dot product property is expressed as:
\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta \]
The result depends on both the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\), and the cosine of the angle (\(\theta\)) between them. When vectors are orthogonal, their dot product is zero because \(\cos 90^\circ = 0\). The property also underpins the Cauchy-Schwarz Inequality, where the absolute value of the dot product cannot exceed the product of the vectors' magnitudes. It's important to understand the dot product because it's widely used in physics, engineering, and computer science to find angles between vectors, project one vector onto another, and more.

Triangle Inequality

The Triangle Inequality is a critical theorem in geometry and vector algebra that stipulates:
\[ |\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}| \]
This means the length of any side of a triangle (represented by the sum of two vectors) is less than or equal to the sum of the lengths of the other two sides. The inequality is a result of the nature of space in Euclidean geometry and it is visually apparent in any physical triangle one might draw. Key to proofs involving the Triangle Inequality is the Cauchy-Schwarz Inequality, as seen in exercises above. The inequality is ubiquitous across mathematics because it defines a fundamental limit on dimensions and measurements within Euclidean space.

Euclidean Space

Euclidean space refers to the two-dimensional and three-dimensional spaces we commonly experience in the physical world, defined by Euclidean geometry. It's a space where points are described by cartesian coordinates, and the distance between points is given by the Euclidean metric.
In Euclidean space, the Triangle Inequality and Cauchy-Schwarz Inequality hold true, reflecting the intuitive concepts of distance and angle. This space forms the foundation for much of classical geometry and vector theory, and it is essential for visualizing and understanding physical phenomena in physics, engineering, and mathematics. Working within Euclidean space involves dealing with vectors, magnitudes, and vector operations like adding vectors and calculating dot products, central to many areas of study.