Question

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u} \| \mathbf{v}|$$

Short answer

Question: Prove the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}| \| |\mathbf{v}|\) and state when the equality holds. Answer: The inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}| \| |\mathbf{v}|\) is proved using the Cauchy-Schwarz inequality, which states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\), their dot product squared is less than or equal to the product of the squared norms of the vectors. The equality holds when the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are linearly dependent, meaning one of them is a scalar multiple of the other. In this case, the vectors are parallel (or anti-parallel) and the dot product equals the product of their norms (with the same or opposite sign).

Step-by-step solution

Recall the Cauchy-Schwarz inequality

Cauchy-Schwarz inequality states that for any vectors \(\mathbf{u}\) and \(\mathbf{v}\), their dot product squared is less than or equal to the product of the squared norms of the vectors: $$(\mathbf{u} \cdot \mathbf{v})^2 \leq |\mathbf{u}|^2 \| |\mathbf{v}|^2$$

Apply the inequality to the given vectors

Using the Cauchy-Schwarz inequality for the vectors \(\mathbf{u}\) and \(\mathbf{v}\), we get: $$(\mathbf{u} \cdot \mathbf{v})^2 \leq |\mathbf{u}|^2 \| |\mathbf{v}|^2$$ We can then take the square root of both sides of the inequality: $$|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u} \| |\mathbf{v}|$$ This proves the desired inequality.

Equality

The equality in the inequality \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u} \| |\mathbf{v}|\) holds if and only if the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are linearly dependent, meaning one of them is a scalar multiple of the other. In this case, the vectors are parallel (or anti-parallel) and the dot product equals the product of their norms (with the same or opposite sign): $$\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| \| |\mathbf{v}|$$

Key concepts

Cauchy-Schwarz Inequality

The Cauchy-Schwarz Inequality is a fundamental inequality in linear algebra, especially in the study of vectors. It essentially provides an upper bound for the absolute value of the dot product of two vectors. Let's break it down to understand it better.

For any two vectors \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \) and \( \mathbf{v} = \langle v_1, v_2, v_3 \rangle \), the Cauchy-Schwarz Inequality states:

\[ (\mathbf{u} \cdot \mathbf{v})^2 \leq (|\mathbf{u}|^2)(|\mathbf{v}|^2) \]

This inequality tells us that the square of the dot product of two vectors is always less than or equal to the product of the squares of their magnitudes. For this reason, it's very helpful in confirming the orthogonality and directionality relations between vectors.

In practical terms, this means that the dot product of two non-zero vectors \( \mathbf{u} \) and \( \mathbf{v} \) has a maximum absolute value of the product of their magnitudes, and it cannot exceed this value. When the vectors are either parallel or anti-parallel, this inequality turns into an equality condition.

Vector Norms

Vector norms, also known as vector magnitudes or lengths, are vital in understanding vector measurements. A norm provides a single positive value that indicates a vector's size or length. Given a vector \( \mathbf{u} = \langle u_1, u_2, u_3 \rangle \), the norm is computed as:

\[ |\mathbf{u}| = \sqrt{u_1^2 + u_2^2 + u_3^2} \]

This formula is akin to the distance formula in three-dimensional space, revealing how far a vector stretches from the origin.

Talking about norms, the most common one encountered in vector calculations is the Euclidean norm, often denoted by \( |\mathbf{u}| \) or \( \| \mathbf{u} \|_2 \). It derives from the famous Pythagorean theorem and is used in many mathematical disciplines.

Understanding norms is crucial as they are used in conjunction with dot products to analyze various vector properties, like angles, distances, and even in defining divergence or convergence in vector sequences. In the context of the Cauchy-Schwarz Inequality, they are pivotal as they define the maximum threshold for the dot product of the vectors.

Linear Dependence

Linear dependence between vectors is an essential concept in vector algebra. It outlines the situation where a vector can be expressed as a scalar multiple of another vector, essentially creating a direct proportional relationship.

In a geometric sense, two vectors \( \mathbf{u} \) and \( \mathbf{v} \) are linearly dependent if you can scale one to get the other. Mathematically, this can be described as:

\[ \mathbf{u} = c \cdot \mathbf{v} \]

Where \( c \) is some scalar constant. This implies that the vectors point in the same direction or exactly opposite directions.

Linear dependence is a necessary and sufficient condition for the Cauchy-Schwarz Inequality to achieve equality. When vectors are linearly dependent, they align precisely, and the dot product of \( \mathbf{u} \) and \( \mathbf{v} \) equals the product of their norms. This specific relationship simplifies many algebraic problems, making the scenarios where it applies significant to recognize. By understanding linear dependence, one can also infer about the span or dimension of the vector space that these vectors constitute.