Question

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. 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

Question: Prove the Triangle Inequality using the Cauchy-Schwarz Inequality, and explain its geometrical interpretation. Answer: To prove the Triangle Inequality, we first expanded the magnitude of the sum of vectors (\( |\mathbf{u}+\mathbf{v}|^2 \)). We then applied the Cauchy-Schwarz Inequality, which states that \( |\mathbf{u}\cdot\mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\), resulting in \( |\mathbf{u}+\mathbf{v}|^2 \leq (|\mathbf{u}| + |\mathbf{v}|)^2 \). By taking the square root of both sides, we proved the Triangle Inequality, which is \(|\mathbf{u}+\mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}|\). Geometrically, the Triangle Inequality can be visualized by considering two vectors \(\mathbf{u}\) and \(\mathbf{v}\) originating from the same point, forming a triangle when added. The inequality states that the length of the third side of the triangle, \(|\mathbf{u}+\mathbf{v}|\), is less than or equal to the sum of the lengths of the other two sides, \(|\mathbf{u}|+|\mathbf{v}|\), which must be true for any triangle.

Step-by-step solution

Expand \(|\mathbf{u}+\mathbf{v}|^{2}\)

To find the magnitude of the sum of vectors \(\mathbf{u}\) and \(\mathbf{v}\), square their magnitude and expand the expression: \begin{align*} |\mathbf{u}+\mathbf{v}|^{2} &= (\mathbf{u}+\mathbf{v}) \cdot (\mathbf{u}+\mathbf{v}) \\ &= \mathbf{u}\cdot\mathbf{u} + \mathbf{u}\cdot\mathbf{v} + \mathbf{v}\cdot\mathbf{u} + \mathbf{v}\cdot\mathbf{v} \\ &= |\mathbf{u}|^{2} + \mathbf{u}\cdot\mathbf{v} + \mathbf{v}\cdot\mathbf{u} + |\mathbf{v}|^{2} \\ &= |\mathbf{u}|^{2} + 2\mathbf{u}\cdot\mathbf{v} + |\mathbf{v}|^{2} \end{align*}

Apply the Cauchy-Schwarz Inequality

From the initial statement, we know that the Cauchy-Schwarz Inequality states that \(|\mathbf{u} \cdot \mathbf{v}|\leq|\mathbf{u}||\mathbf{v}|\). Since \(\mathbf{u}\cdot\mathbf{v}\) is a scalar, the inequality can be rewritten as: \begin{equation*} 2\mathbf{u}\cdot\mathbf{v} \leq 2|\mathbf{u}||\mathbf{v}| \end{equation*} Now use this inequality, and substitute it in the expression from Step 1: \begin{align*} |\mathbf{u}+\mathbf{v}|^{2} &\leq |\mathbf{u}|^{2} + 2|\mathbf{u}||\mathbf{v}| + |\mathbf{v}|^{2} \\ &= (|\mathbf{u}|+|\mathbf{v}|)^{2} \end{align*}

Prove the Triangle Inequality

We have shown that \(|\mathbf{u}+\mathbf{v}|^2 \leq (|\mathbf{u}| + |\mathbf{v}|)^2\). To finally prove the Triangle Inequality, take the square root of both sides: \begin{equation*} |\mathbf{u}+\mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}| \end{equation*} And thus, the Triangle Inequality has been proven.

Geometrical Interpretation

The geometrical interpretation of the Triangle Inequality in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) can be visualized as follows: Consider two vectors \(\mathbf{u}\) and \(\mathbf{v}\) originating from the same point (the origin). When these two vectors are added, they form a triangle, with \(\mathbf{u}+\mathbf{v}\) being the third side of the triangle. The Triangle Inequality states that the length of the third side (\(|\mathbf{u}+\mathbf{v}|\)) is less than or equal to the sum of lengths of the other two sides (\(|\mathbf{u}|+|\mathbf{v}|\)), which must be true for any triangle.

Key concepts

Triangle Inequality

The Triangle Inequality is a fundamental concept in geometry and vector algebra. It states that for any two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the magnitude of their sum is less than or equal to the sum of their magnitudes:

• Mathematically, this is expressed as \( |\mathbf{u} + \mathbf{v}| \leq |\mathbf{u}| + |\mathbf{v}| \).
• This inequality holds in any dimensional space, making it a powerful tool in vector analysis.

The Triangle Inequality is visually intuitive. Picture two vectors as two sides of a triangle, while the third side is their resultant vector. This vision emphasizes that the shortest path between two points is a straight line, which is less than or equal to taking the sum of two indirect paths. By satisfying this mathematical property, the inequality ensures the rationality of vector addition in the spatial domain.

Vector Magnitude

Understanding vector magnitude is crucial when working with vectors. The magnitude (or length) of a vector \( \mathbf{u} \) is denoted as \( |\mathbf{u}| \):

• It represents the distance from the origin to the endpoint of the vector in a coordinate space.
• The magnitude can be calculated using the Euclidean formula, \( |\mathbf{u}| = \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2} \), where \( x_1, x_2, \ldots, x_n \) are the components of \( \mathbf{u} \).

The concept of magnitude is integral in vector operations, including verifying the Triangle Inequality. When computing \( |\mathbf{u} + \mathbf{v}| \), both individual magnitudes of \( \mathbf{u} \) and \( \mathbf{v} \) play critical roles. The resultant magnitude is crucial for understanding comparisons in vector space, determining the length of vector sums, and applying the Triangle Inequality.

Geometric Interpretation

The geometric interpretation of vector operations enhances comprehension of theoretical principles. The Triangle Inequality, when extended to geometry, forms a visual narrative:

• Envision two vectors \( \mathbf{u} \) and \( \mathbf{v} \) plotted in a plane, originating together at a single point.
• Their sum, \( \mathbf{u} + \mathbf{v} \), forms the diagonal of a parallelogram or the third side of a triangle meant by the vectors head-to-tail.

In more practical terms, consider navigating a path within a square where two adjoining sides represent the vectors. By traversing the diagonal instead, you take the shortest route, highlighting why the Triangle Inequality is always valid. This visual analogy simplifies complex algebraic inequalities by translating them into intuitive geometric relationships, clarifying the logic behind vector sums and inequalities.

Dot Product

The dot product is a crucial vector operation connecting algebra and geometry. Given two vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product, expressed as \( \mathbf{u} \cdot \mathbf{v} \), encapsulates mutual directional influence:

• It is computed as \( u_1v_1 + u_2v_2 + \cdots + u_nv_n \), involving component-wise multiplication and summation.
• Geometrically, it correlates with the cosine of the angle \( \theta \) between the two vectors: \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta \).

The dot product helps manifest the Cauchy-Schwarz Inequality. Since it links to the angle between vectors, the absolute value of the dot product is bounded by the product of their magnitudes. Recognizing this bounds comprehension of vector relationships and influences the understanding of the Triangle Inequality.