Problem 51 - Advanced Engineering Mathematics

Question

Use the dot product to prove the triangle inequality \(\|\mathbf{a}+\mathbf{b}\| \leq\|\mathbf{a}\|+\|\mathbf{b}\| .\) [Hint: Consider property \((v i)\) of the dot product.]

Step-by-Step Solution

Answer

The triangle inequality is proven by applying the dot product distributive property and comparing the squares.

Step 1: Recall the Dot Product Properties

The dot product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta) \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \). Furthermore, the dot product has the non-negativity property stating \( \| \mathbf{a} \|^2 \geq 0 \). These will be important for our proof.

Step 2: Apply the Dot Product to the Sum

For two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the norm of their sum \( \| \mathbf{a} + \mathbf{b} \|^2 \) can be expressed using the dot product as follows: \( \| \mathbf{a} + \mathbf{b} \|^2 = (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) \). Using the distributive property, this equals \( \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} \).

Step 3: Express in Terms of Magnitudes

Substituting for the dot products, we find: \( \| \mathbf{a} + \mathbf{b} \|^2 = \| \mathbf{a} \|^2 + 2 \mathbf{a} \cdot \mathbf{b} + \| \mathbf{b} \|^2 \). Since \( \mathbf{a} \cdot \mathbf{b} \leq \| \mathbf{a} \| \| \mathbf{b} \| \), we have that \( 2 \mathbf{a} \cdot \mathbf{b} \leq 2 \| \mathbf{a} \| \| \mathbf{b} \| \).

Step 4: Compare Squares to Derive Inequality

Thus, \( \| \mathbf{a} + \mathbf{b} \|^2 \leq \| \mathbf{a} \|^2 + 2 \| \mathbf{a} \| \| \mathbf{b} \| + \| \mathbf{b} \|^2 \). Recognizing the right side as \((\|\mathbf{a}\| + \|\mathbf{b}\|)^2\), we have \( \| \mathbf{a} + \mathbf{b} \|^2 \leq (\|\mathbf{a}\| + \|\mathbf{b}\|)^2 \).

Step 5: Take Square Roots

Taking the square root of both sides, we conclude \( \| \mathbf{a} + \mathbf{b} \| \leq \| \mathbf{a} \| + \| \mathbf{b} \| \), which completes the proof of the triangle inequality.

Key Concepts

Dot Product

The dot product is a mathematical operation that takes two vectors and returns a single scalar value. It is calculated by multiplying the corresponding components of the vectors and then summing these products. Specifically, for vectors \( \mathbf{a} \) and \( \mathbf{b} \), the dot product is given by \( \mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos(\theta) \), where \( \theta \) is the angle between the vectors.
This operation has several important properties:

• Commutativity: \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \)
• Distributivity: \( \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \)
• Non-Negativity: \( \mathbf{a} \cdot \mathbf{a} \geq 0 \), meaning the dot product of a vector with itself is always non-negative.

These properties make the dot product a powerful tool in vector mathematics, particularly when analyzing geometric figures and proving inequalities like the triangle inequality.

Vector Norm

The vector norm, often referred to as the magnitude or length of a vector, is a measure of its size. For a vector \( \mathbf{a} \), the norm is denoted as \( \|\mathbf{a}\| \) and is calculated using the dot product: \( \|\mathbf{a}\| = \sqrt{\mathbf{a} \cdot \mathbf{a}} \).
The vector norm comes with crucial properties:

• Non-Negativity: \( \|\mathbf{a}\| \geq 0 \)
• Zero Vector: \( \|\mathbf{a}\| = 0 \) if and only if \( \mathbf{a} \) is the zero vector.
• Scalar Multiplication: \( \|c\mathbf{a}\| = |c| \cdot \|\mathbf{a}\| \) for any scalar \( c \).

This concept plays a pivotal role in establishing the triangle inequality, as it sets a foundation for comparing different vectors.

Mathematical Proof

A mathematical proof is a step-by-step demonstration of the truth of a mathematical statement. Proofs are essential in mathematics because they provide a logical framework for verifying that statements are indeed true under the assumptions set by the problem.
The structure of a proof generally involves:

• Assumptions: Given or known information that can be used.
• Logical Steps: Each step follows logically from the previous one, often employing mathematical properties and theorems.
• Conclusion: The final step, where the initial statement is shown to be true.

Proofs can be direct, like the one used in the demonstration of the triangle inequality, or they might employ other strategies such as contradiction or induction. They are fundamental in validating mathematical concepts.

Triangle Inequality Proof

The triangle inequality is a fundamental concept in geometry and vector mathematics, stating that for any vectors \( \mathbf{a} \) and \( \mathbf{b} \), the sum of their magnitudes is at least as great as the magnitude of their sum: \( \|\mathbf{a} + \mathbf{b}\| \leq \|\mathbf{a}\| + \|\mathbf{b}\| \).
To prove this using the dot product, follow these steps:

• Recognize that \( \|\mathbf{a} + \mathbf{b}\|^2 = (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) \).
• Expand this expression using distributive property: \( \mathbf{a} \cdot \mathbf{a} + 2 \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b} \).
• Relate it to known magnitudes and use the fact that \( \mathbf{a} \cdot \mathbf{b} \leq \|\mathbf{a}\| \|\mathbf{b}\| \).
• Conclude by establishing that \( \|\mathbf{a} + \mathbf{b}\|^2 \leq (\|\mathbf{a}\| + \|\mathbf{b}\|)^2 \).
• Take square roots to complete the inequality.

This proof showcases not only the elegance of mathematical reasoning but also the practical use of the dot product and vector norms in establishing geometric truths.