Problem 50 - Essential Calculus Early Transcendentals

Question

The Triangle Inequality for vectors is $$|\mathbf{a}+\mathbf{b}| \leqslant|\mathbf{a}|+|\mathbf{b}|$$ (a) Give a geometric interpretation of the Triangle Inequality. (b) Use the Cauchy-Schwarz Inequality from Exercise 49 to prove the Triangle Inequality. [Hint: Use the fact that \(|\mathbf{a}+\mathbf{b}|^{2}=(\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}+\mathbf{b})\) and use Property 3 of the dot product.]

Step-by-Step Solution

Answer

(a) It means the triangle's third side is never longer than the sum of the other two. (b) Use the dot product and Cauchy-Schwarz Inequality for the proof.

Step 1: Understand the Geometric Interpretation

The Triangle Inequality for vectors states that the magnitude (length) of the sum of two vectors \(\mathbf{a} + \mathbf{b}\) is always less than or equal to the sum of the magnitudes of the two vectors \(|\mathbf{a}| + |\mathbf{b}|\). Geometrically, this can be visualized with vectors originating from the same point forming a triangle with their resultant, \(\mathbf{a} + \mathbf{b}\), as the third side. According to the inequality, this resultant vector's length will never exceed the length obtained by simply following the path along the two vectors.

Step 2: Express the Magnitude Squared in Terms of Dot Product

We begin proving the Triangle Inequality by using the dot product. Remember that the square of the magnitude of a vector \(\mathbf{c}\) is given by \(|\mathbf{c}|^2 = \mathbf{c} \cdot \mathbf{c}\). Applying this to \(\mathbf{a} + \mathbf{b}\), we have:\[|\mathbf{a} + \mathbf{b}|^2 = (\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b})\].

Step 3: Expand Using Dot Product Properties

Use the distributive property of the dot product:\[(\mathbf{a} + \mathbf{b}) \cdot (\mathbf{a} + \mathbf{b}) = \mathbf{a} \cdot \mathbf{a} + 2\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{b}\].Thus, we can express it as:\[|\mathbf{a} + \mathbf{b}|^2 = |\mathbf{a}|^2 + 2\mathbf{a}\cdot\mathbf{b} + |\mathbf{b}|^2\].

Step 4: Apply the Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality states that \[(\mathbf{a} \cdot \mathbf{b})^2 \leq |\mathbf{a}|^2 |\mathbf{b}|^2\].Taking the square root of both sides and considering the possible sign of \(\mathbf{a} \cdot \mathbf{b}\),\(|\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}||\mathbf{b}|\). Therefore:\[2\mathbf{a} \cdot \mathbf{b} \leq 2|\mathbf{a}||\mathbf{b}|\].

Step 5: Conclude the Proof

Substitute the inequality from Step 4 into the expanded expression from Step 3:\[|\mathbf{a} + \mathbf{b}|^2 \leq |\mathbf{a}|^2 + 2|\mathbf{a}||\mathbf{b}| + |\mathbf{b}|^2\].Recognizing the right side as a perfect square: \[(|\mathbf{a}| + |\mathbf{b}|)^2\], we can thus write:\[|\mathbf{a} + \mathbf{b}|^2 \leq (|\mathbf{a}| + |\mathbf{b}|)^2\].Taking the square root of both sides gives:\[|\mathbf{a} + \mathbf{b}| \leq |\mathbf{a}| + |\mathbf{b}|\].This completes the proof of the Triangle Inequality for vectors.

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude and direction. They can be visualized as arrows in space pointing from one point to another. Each vector has specific properties:

• Magnitude (length): It represents the size or length of the vector and is always a non-negative value.
• Direction: This indicates the path along which the vector acts, demonstrating where it points in space.

In physics and engineering, vectors are crucial for describing phenomena such as velocity, force, and more. When dealing with vectors graphically, they are often represented by directed line segments, with the initial point being the tail and the terminal point being the head.

Operations involving vectors include addition, subtraction, scaling (changing magnitude), and finding resultant vectors. For instance, when adding vectors, you pull them tail to head to form a single new vector that represents the cumulative effect of the original vectors.

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a fundamental principle in mathematics, particularly in linear algebra and analysis. It provides a crucial inequality between the dot product of two vectors and the product of their magnitudes:

• It states that for any vectors \( \mathbf{a} \) and \( \mathbf{b} \), the square of their dot product is less than or equal to the product of their magnitudes.

In formulaic terms: \[ (\mathbf{a} \cdot \mathbf{b})^2 \leq |\mathbf{a}|^2 |\mathbf{b}|^2 \]This inequality implies that the angle between the vectors causes the dot product to always be constrained by the sizes of the vectors.

It is valuable in proofs and derivations in various mathematical contexts like in the Triangle Inequality, because it provides a bridge between algebraic expressions (dot products) and geometric understandings (magnitudes of vectors).

Understanding the Cauchy-Schwarz Inequality deepens your comprehension of one of the qualitative features of vector spaces, often leading to insights about orthogonality and projections in more complex mathematical studies.

Dot Product

The dot product, also known as the scalar product, is an algebraic operation that takes two equal-length sequences of numbers (usually coordinates of vectors) and returns a single number. It is called a dot product because of the dot symbol \((\cdot)\) used in identifying it. Some key traits of the dot product are:

• Computation: For vectors \(\mathbf{a} = (a_1, a_2, ..., a_n)\) and \(\mathbf{b} = (b_1, b_2, ..., b_n)\), the dot product is computed as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n \]
• Result: The dot product results in a scalar, unlike vector products which produce vectors.

The dot product is significant for multiple reasons: it helps determine the angle between vectors, assess orthogonality (vectors with a dot product of zero are perpendicular), and find projections of vectors in physical interpretations.

It plays an integral role along with Cauchy-Schwarz in proofs that involve inequalities, such as the Triangle Inequality, due to its ability to quantify aspects of vectors into a single, insightful value.

Geometric Interpretation

In geometry, the Triangle Inequality gives a meaningful visualization of how vectors interact in space. Imagine two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), originating from the same point and forming two sides of a triangle with their resultant vector, \( \mathbf{a} + \mathbf{b} \), completing the third side.

The Triangle Inequality states:\[ |\mathbf{a} + \mathbf{b}| \leq |\mathbf{a}| + |\mathbf{b}| \]Geometrically, this implies that the direct path (the vector resultant) between two points is always the shortest path when compared to any detour path (the two separate vectors). You can visualize this by imagining it as being more efficient to walk directly from a point to another rather than taking a longer pre-defined route.

This interpretation helps provide intuition behind the inequality, showing us not only an algebraic truth but also a practical principle of navigation and travel in vector spaces, which is not only true in mathematics but also in real-world applications of geometry.