Question

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

Short answer

Answer: Equality in the Cauchy-Schwarz Inequality holds when the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel to each other, either pointing in the same or opposite directions. This occurs when the two vectors are scalar multiples of each other, that is, \(\mathbf{u} = k\mathbf{v}\) or \(\mathbf{v} = k\mathbf{u}\), where \(k\) is a scalar value.

Step-by-step solution

Understanding the Dot Product Definition

Recall the definition of the dot product, which is given as \(\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta\), where \(\theta\) is the angle between the two vectors. The Cauchy-Schwarz Inequality is derived from this definition and states that \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\).

Determine the Conditions for Equality

We want to find the conditions that lead to equality in the inequality, that is, \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\). From the definition of the dot product, equality holds when \(|\cos \theta| = 1\). This is only true when \(\cos \theta = \pm 1\).

Analyze the Cases for \(\cos{\theta}\)

There are two cases to consider for \(\cos (\theta)\): 1. \(\cos (\theta) = 1\) implies that the angle between the two vectors is 0 degrees, meaning that the vectors are parallel and point in the same direction. 2. \(\cos (\theta) = -1\) implies that the angle between the two vectors is 180 degrees, meaning that the vectors are parallel and point in opposite directions.

Conclusion

The conditions for equality in the Cauchy-Schwarz Inequality are satisfied when the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are parallel to each other (point in the same or opposite directions). In other words, when the two vectors are scalar multiples of each other, that is, \(\mathbf{u} = k\mathbf{v}\) or \(\mathbf{v} = k\mathbf{u}\), where \(k\) is a scalar value.

Key concepts

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It is denoted and calculated as \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta \), where \(\theta\) represents the angle between vectors \(\mathbf{u}\) and \(\mathbf{v}\). This calculation plays a crucial role in understanding the geometrical and physical significance of vector multiplication.

The dot product reflects the degree to which one vector extends in the same direction as another. It is fundamentally related to the concepts of vector projection and work in physics. For instance, if the vectors are perpendicular, the dot product is zero because \( \cos 90^\circ = 0 \), indicating no magnitude in the common direction of the vectors. When the vectors are parallel, either pointing in the same or opposite directions, the dot product reaches its maximum or minimum value, respectively, because \( \cos 0^\circ = 1 \) and \( \cos 180^\circ = -1 \).

By understanding the dot product, one can solve for angles between vectors, determine vector lengths when combined, and work with the principles underpinning the Cauchy-Schwarz Inequality.

Vector Magnitude

The magnitude of a vector is the 'length' of the vector from its starting point to its end point in a vector space. It is denoted as \( |\mathbf{u}| \) for vector \( \mathbf{u} \) and is calculated by taking the square root of the sum of the squared components of the vector.

Written explicitly for a vector \( \mathbf{u} = (u_1, u_2, ..., u_n) \), the magnitude is \( |\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2} \). In the context of the Cauchy-Schwarz Inequality, which involves the dot product, the magnitudes of the vectors play a critical role. Since the inequality states \( |\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}| \), understanding how to calculate vector magnitudes is essential.

Magnitudes allow us to measure the 'size' of vectors irrespective of their direction and are also necessary to determine when the Cauchy-Schwarz Inequality reaches equality—that is when the magnitude of the dot product equals the product of vector magnitudes.

Vector Parallelism

Vector parallelism refers to the condition when two vectors have the same or exactly opposite direction in space. For vectors to be parallel, it is necessary that one vector can be expressed as a scalar multiple of the other. This means that the two vectors are proportional to each other in their components.

Mathematically, if vectors \( \mathbf{u} \) and \( \mathbf{v} \) are parallel, then there exists a scalar \( k \) such that \( \mathbf{u} = k\mathbf{v} \) or conversely. When considering the Cauchy-Schwarz Inequality, parallelism is the condition that leads to equality because the vectors' directions are either exactly aligned (\( \cos \theta = 1 \) when \( k \) is positive) or diametrically opposed (\( \cos \theta = -1 \) when \( k \) is negative).

Recognizing parallel vectors is an essential skill in vector analysis and has practical applications in physics, engineering, and computer graphics. It allows for the simplification of complex systems into more manageable calculations by understanding the underlying geometric relationships between system components.

Scalar Multiples

A scalar multiple of a vector involves multiplying each component of the vector by a single number, known as a scalar. If \( \mathbf{v} \) is a vector and \( k \) is a scalar, the scalar multiple is \( k\mathbf{v} \) which is a new vector. The properties of scalar multiplication include that the resulting vector \( k\mathbf{v} \) has the same direction as \( \mathbf{v} \) if \( k > 0 \) and the opposite direction if \( k < 0 \) but has a magnitude \( |k| \) times the magnitude of \( \mathbf{v} \).

Scalar multiples are significant when discussing the Cauchy-Schwarz Inequality where vectors \( \mathbf{u} \) and \( \mathbf{v} \) reach equality if and only if one is a scalar multiple of the other. This relationship is a powerful tool for deducing properties about vectors, such as parallelism, and for performing geometric transformations like scaling. Understanding scalar multiples is vital in many areas of mathematics and physics, like workings involving vector fields and wave functions.