Problem 29 - Thomas Calculus

Question

a. Cauchy-Schwartz inequality Use the fact that \(\mathbf{u} \cdot \mathbf{v}=\) \(|\mathbf{u}||\mathbf{v}| \cos \theta\) to 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

The inequality holds always; equality occurs if vectors are parallel.

Step 1: Understanding the Dot Product

The dot product formula for two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta\), where \(|\mathbf{u}|\) and \(|\mathbf{v}|\) are the magnitudes of \(\mathbf{u}\) and \(\mathbf{v}\) respectively, and \(\theta\) is the angle between them.

Step 2: Expressing the Inequality

We need to show \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\). We begin by substituting the expression for the dot product: \(|\mathbf{u} \cdot \mathbf{v}| = |\,|\mathbf{u}||\mathbf{v}| \cos \theta\,|\).

Step 3: Analyzing the Cosine Function

The cosine function has a range of \([-1, 1]\). Thus, the value of \(\cos \theta\) can vary between \(-1\) and \(1\), making \(|\cos \theta| \leq 1\).

Step 4: Applying the Range of Cosine

Since \(|\cos \theta| \leq 1\), we have:\[|\,|\mathbf{u}||\mathbf{v}| \cos \theta\,| = |\mathbf{u}||\mathbf{v}| \cdot |\cos \theta| \leq |\mathbf{u}||\mathbf{v}|\] This confirms the Cauchy-Schwartz inequality.

Step 5: Conditions for Equality

The equality \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\) holds if and only if \(|\cos \theta| = 1\). This occurs when \(\cos \theta = 1\) or \(\cos \theta = -1\), which means \(\theta = 0^\circ\) or \(\theta = 180^\circ\).

Step 6: Interpreting the Condition for Equality

When \(\theta = 0^\circ\), vectors \(\mathbf{u}\) and \(\mathbf{v}\) are pointing in the same direction. When \(\theta = 180^\circ\), they are pointing in opposite directions. In both cases, the vectors are parallel. Thus, the equality holds when \(\mathbf{u}\) and \(\mathbf{v}\) are parallel.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra, particularly when working with vectors in a Euclidean space. It is defined as the product of the magnitudes of two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), and the cosine of the angle \(\theta\) between them: \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta\). This operation results in a scalar quantity, hence the alternative name "scalar product." The usefulness of the dot product lies in its ability to quantify how much one vector "goes in the direction" of another. If \(\cos \theta = 0\), it signifies that the vectors are perpendicular, meaning they have no component in each other's direction.

Vector Magnitude

The magnitude of a vector is a measure of its "length" or "size." For a vector \(\mathbf{u} = (u_1, u_2, ..., u_n)\), the magnitude is given by the formula: \[|\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2} \]

• The magnitude is always a non-negative number, i.e., \(|\mathbf{u}| \geq 0\).
• For a zero vector, the magnitude is zero: \(|\mathbf{0}| = 0\).
• The magnitude helps in identifying the length and direction of the vector, essential for defining norms in vector spaces.

Understanding vector magnitude is crucial for calculating the dot product and analyzing vector behavior in equations.

Angle Between Vectors

The angle between vectors determines the dot product's behavior. The formula \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta\) includes \(\cos \theta\), which is the cosine of the angle between the vectors. The angle reveals important relational characteristics:

• If \(\theta = 0^\circ\), the vectors are aligned or pointing in exactly the same direction.
• If \(\theta = 90^\circ\), the cosine is zero, indicating perpendicular vectors.
• If \(\theta = 180^\circ\), \(\cos \theta = -1\), showing the vectors are in opposite directions.

By understanding these angle properties, the connection between vectors can be interpreted geometrically and algebraically.

Parallel Vectors

Parallel vectors either point in the same direction or directly opposite to each other. This occurs at specific angles between vectors:

• When \(\theta = 0^\circ\), vectors \(\mathbf{u}\) and \(\mathbf{v}\) point in the same direction.
• When \(\theta = 180^\circ\), they point in opposite directions.
• For both cases, \(\cos \theta = \pm 1\), showing maximum or minimum alignment.

The Cauchy-Schwartz inequality confirms parallelism when \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\), meaning no loss in magnitude for the dot product compared to the product of the magnitudes. When you identify vectors are parallel, you can solve many geometric and physical problems by making projections and analyzing directions straightforwardly.