Problem 27 - University Calculus: Early Transcendentals

Question

a. Cauchy-Schwarz inequality since \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) show that the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|\) holds for any vectors \(\mathbf{u}\) and \(\mathbf{v}\). b. Under what circumstances, if any, does \(|\mathbf{u} \cdot \mathbf{v}|\) equal \(|\mathbf{u}||\mathbf{v}| ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

a. The inequality holds due to \(|\cos \theta| \leq 1\). b. Equality occurs when vectors are parallel.

Step 1: Understand the Dot Product Formula

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is given by \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta\), where \(\theta\) is the angle between the vectors. This formula helps in linking the geometric interpretation of dot product with its algebraic computation.

Step 2: State the Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\), the absolute value of their dot product is less than or equal to the product of their magnitudes: \[|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|.\]

Step 3: Apply Geometry to Understand the Inequality

Since \(|\cos \theta| \leq 1\) for all angles \(\theta\), it follows that \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}| |\cos \theta| \leq |\mathbf{u}||\mathbf{v}|\). This demonstrates that the inequality holds.

Step 4: Identify When Equality Holds

Equality \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\) holds when \(|\cos \theta| = 1\). This occurs when \(\theta = 0\degree\) or \(\theta = 180\degree\), meaning \(\mathbf{u}\) and \(\mathbf{v}\) are parallel (same or opposite direction).

Key Concepts

Dot Product

The dot product is an essential operation when dealing with vectors in mathematics and physics. For two vectors, say \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \), the dot product is algebraically calculated as \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). But there's more than just the numbers!

On a geometric level, the dot product also represents the product of the magnitudes of the vectors and the cosine of the angle \( \theta \) between them: \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta \).

This means that the dot product is connected to both the vectors' directions and lengths.

• If the dot product is positive, the angle \( \theta \) is acute.
• If it is zero, the vectors are orthogonal (90 degrees).
• If negative, \( \theta \) is obtuse.

This insight helps to identify whether vectors point in roughly the same direction, opposite directions, or perpendicular directions.

Vector Magnitude

A vector's magnitude, sometimes called its length or norm, is a measure of how 'long' the vector is. It is calculated using the square root of the sum of the vector's components squared. For a vector \( \mathbf{u} = (u_1, u_2) \), the magnitude is \( |\mathbf{u}| = \sqrt{u_1^2 + u_2^2} \).

Magnitude serves as an essential part of understanding vector properties and operations. Specifically, the magnitude is crucial in the context of the Cauchy-Schwarz inequality, where it helps compare the length component across vectors.

When working with the dot product, the magnitude scales the vector product. In simpler terms, if two vectors differ only by length but point in the same direction, their dot product will naturally increase just because their magnitude is larger.

Thus, understanding magnitudes provides a deeper insight into comparing and operating with vectors, especially in considering distances, lengths, and interpretations of vector space.

Geometric Interpretation

The Cauchy-Schwarz inequality is not only a fundamental algebraic tool but also has a tangible geometric interpretation. Busy visualizing vectors in space—think of them as arrows pointing outwards. The inequality tells us about the relationship between these arrows.

Geometrically, the Cauchy-Schwarz inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\) ensures that the dot product magnitude cannot exceed the product of the individual vector magnitudes, limited by the maximum value of \( |\cos \theta| \leq 1 \).

In simple terms:

• If \(\theta = 0\degree\), the vectors are perfectly aligned, thus \(|\cos \theta| = 1\) and the inequality becomes an equality \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\).
• If \(\theta\) is something else, the dot product is smaller due to multiplicative effect of \( \cos(\theta) \) reducing the impactful 'stretch'.

Thus, the geometric perspective underlines how alignment and directionality between vectors influence calculations, which is key in fields ranging from physics to engineering.