Question

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) 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 Cauchy-Schwarz Inequality?

Short answer

Answer: The Cauchy-Schwarz Inequality becomes an equality for two vectors when they are scalar multiples of each other, meaning they are collinear and point in the same or opposite directions.

Step-by-step solution

Understand the dot product formula

The dot product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is defined as \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\), where \(\theta\) is the angle between the vectors. From this, we have the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}|\) (because \(|\cos \theta| \leq 1\)).

Examine equality condition for cosine

The only way for the inequality to become an equality is if \(|\cos \theta| = 1\). As cosine function values can only be between -1 and 1, \(|\cos \theta| = 1\) means that the angle between the vectors can only be one of the two angles: \(\theta = 0\) (cosine of 0 is 1) or \(\theta = \pi\) (cosine of \(\pi\) is -1).

Describe the geometric conditions for equality

If the angle between the two vectors is \(0\), then the vectors are collinear and point in the same direction. If the angle between the two vectors is \(\pi\), then the vectors point in opposite directions but are still collinear. In both cases, the vectors are scalar multiples of each other, meaning \(\mathbf{u} = c\mathbf{v}\), where \(c\) is a scalar.

Summarize the conditions for equality

In conclusion, the conditions on \(\mathbf{u}\) and \(\mathbf{v}\) that lead to equality in the Cauchy-Schwarz Inequality are when \(\mathbf{u}\) and \(\mathbf{v}\) are scalar multiples of each other. This happens when the two vectors are collinear and point in the same or opposite directions.

Key concepts

Dot Product

The dot product is a crucial concept in vector calculus. It is a way to multiply two vectors that produces a scalar, rather than another vector. For two vectors \(\mathbf{u}\) and \(\mathbf{v}\), the dot product is calculated using the formula: \[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos \theta \] where \(\theta\) is the angle between the vectors. This formula ties together both the magnitudes of the vectors and the cosine of the angle between them.

• If \(\cos \theta = 1\), the vectors are pointing in the same direction.
• If \(\cos \theta = -1\), they point in opposite directions.
• If \(\cos \theta = 0\), the vectors are orthogonal (perpendicular).

Understanding the dot product is essential for grasping other vector calculus concepts, like the Cauchy-Schwarz Inequality.

Vector Calculus

Vector calculus provides the foundation for analyzing and tackling problems that involve vectors, especially in physics and engineering. It deals with vector fields and operations on vectors, such as differentiation and integration of vector fields.

• A vector in this context is a quantity with both magnitude and direction.
• Key operations include dot product, cross product, and scalar multiplication.

The dot product is often used to determine angles between vectors, projections, and for simplifying calculations involving complex vector expressions. Through vector calculus, we can model and solve real-world phenomena, including forces, fluid flow, and electromagnetic fields.

Geometric Interpretation

The geometric interpretation of the dot product is rooted in the relationship between the magnitudes of vectors and the cosine of the angle between them. Simply put, it measures how much one vector extends in the direction of another vector. This is why the dot product is zero when the vectors are perpendicular, as there is no extension in each other's direction. When we talk about the conditions for equality in the Cauchy-Schwarz Inequality, we're looking at scenarios where the two vectors are collinear. This means:

• The vectors are parallel, potentially pointing in the same or opposite directions.
• The relationship \(\mathbf{u} = c\mathbf{v}\) signifies that one vector is a scalar multiple of the other.

In geometric terms, this scalar multiplication indicates the vectors overlap or reflect along the same line.

Scalar Multiplication

Scalar multiplication involves scaling a vector by a constant number, called a scalar. This simple yet powerful operation can explain why the Cauchy-Schwarz Inequality holds as an equality under certain conditions. When you have two vectors \(\mathbf{u}\) and \(\mathbf{v}\), and one is a scalar multiple of the other (\(\mathbf{u} = c\mathbf{v}\)), their dot product essentially boils down to scaling.

• If \(c > 0\), the vectors have the same direction.
• If \(c < 0\), they have opposite directions.
• If \(c = 1\), they are identical in direction and magnitude.

Thus, scalar multiplication provides a straightforward way to assess how vectors relate to each other in terms of direction and proportionality, which is crucial in understanding the Cauchy-Schwarz Inequality conditions.