Question

a. Show that the Cauchy-Schwarz inequality \(\mid \mathbf{u}\). \(\mathbf{v} \mid \leq\|\mathbf{u}\|\|\mathbf{v}\|\) holds for all vectors \(\mathbf{u}\) and \(\mathbf{v}\). [Hint: \(|\cos \theta| \leq 1\) for all angles \(\theta .]\) b. Show that \(|\mathbf{u} \cdot \mathbf{v}|=\|\mathbf{u}\|\|\mathbf{v}\|\) if and only if \(\mathbf{u}\) and \(\mathbf{v}\) are parallel. [Hint: When is \(\cos \theta=\pm 1 ?]\) c. Show that \(\left|x_{1} x_{2}+y_{1} y_{2}+z_{1} z_{2}\right|\) \(\leq \sqrt{x_{1}^{2}+y_{1}^{2}+z_{1}^{2}} \sqrt{x_{2}^{2}+y_{2}^{2}+z_{2}^{2}}\) holds for all numbers \(x_{1}, x_{2}, y_{1}, y_{2}, z_{1},\) and \(z_{2}\) d. Show that \(|x y+y z+z x| \leq x^{2}+y^{2}+z^{2}\) for all \(x, y\), and \(z\) e. Show that \((x+y+z)^{2} \leq 3\left(x^{2}+y^{2}+z^{2}\right)\) holds for all \(x, y,\) and \(z\)

Short answer

All parts utilize Cauchy-Schwarz or related inequalities, showing vector/magnitude relationships or using quadratic expansions.

Step-by-step solution

Understand the Cauchy-Schwarz inequality

The Cauchy-Schwarz inequality states that for any vectors \( \mathbf{u} \) and \( \mathbf{v} \), \( |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\| \). This can be understood in the context of the dot product and the norms (magnitudes) of these vectors. The dot product expresses how much one vector goes in the direction of the other.

Prove Part a

Consider the geometric interpretation of the dot product: \( \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta \), where \( \theta \) is the angle between the two vectors. Using the hint, we note that \(|\cos \theta| \leq 1\) for all angles \( \theta \). Therefore, applying this inequality gives \( |\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\| \cdot 1 = \|\mathbf{u}\| \|\mathbf{v}\|\).

Prove Part b

The condition \(|\mathbf{u} \cdot \mathbf{v}| = \|\mathbf{u}\| \|\mathbf{v}\| \) implies \(\cos \theta = \pm 1\). This happens only when \(\theta = 0 \text{ or } \pi\), meaning \( \mathbf{u} \text{ and } \mathbf{v} \) are parallel (same or opposite direction).

Prove Part c

Part c involves vectors in three dimensions. Let \(\mathbf{u} = (x_1, y_1, z_1)\) and \(\mathbf{v} = (x_2, y_2, z_2)\). The dot product is \(\mathbf{u} \cdot \mathbf{v} = x_1 x_2 + y_1 y_2 + z_1 z_2\), and the Cauchy-Schwarz inequality becomes \(|\mathbf{u} \cdot \mathbf{v}| \leq \|\mathbf{u}\| \|\mathbf{v}\|\). This reaffirms the inequality described, as it's the same as Cauchy-Schwarz.

Prove Part d using the AM-GM inequality

We rewrite \(|xy + yz + zx| \leq x^2 + y^2 + z^2\). By Cauchy-Schwarz in disguise, if we consider any three positive terms, the inequality holds; each term can be seen maximizing its value as a square expression, thus \(3t(u^2 + v^2 + w^2)\) dominates over cross terms.

Prove Part e using quadratic expansion

Expand \((x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx\). Using AM-GM and realizing they've symmetric cross terms being the likely weaker part, add squared terms to ensure all components as required part towards \(3(x^2 + y^2 + z^2)\).

Key concepts

Dot Product

The dot product is a way to multiply two vectors and get a scalar (a single number). It is denoted as \( \mathbf{u} \cdot \mathbf{v} \) when dealing with vectors \( \mathbf{u} \) and \( \mathbf{v} \). This operation looks like this in detail: the dot product of two vectors \((a_1, a_2, a_3)\) and \((b_1, b_2, b_3)\) is calculated by:
\[ a_1b_1 + a_2b_2 + a_3b_3 \]The result you get from this is a number, not another vector.

• It describes how much one vector extends in the direction of another vector.
• If the vectors point in the same direction, the dot product is large.
• If they are perpendicular, the dot product is zero, indicating no extension in those directions.

Understanding the dot product helps in many mathematical scenarios, especially because of its role in vector spaces and its usefulness in calculating angles between vectors.

Geometric Interpretation

Geometrically, the dot product represents a relationship between two vectors regarding their directions and angles. According to the formula:
\[ \mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta \]where \( \|\mathbf{u}\| \) and \( \|\mathbf{v}\| \) are the norms (or lengths) of the vectors, and \( \theta \) is the angle between them.

• If \( \theta = 0 \), the vectors are parallel and point in the same direction, making \( \cos \theta = 1 \).
• If \( \theta = \pi \), the vectors are parallel but point in opposite directions, making \( \cos \theta = -1 \).
• If \( \theta = \frac{\pi}{2} \), the vectors are perpendicular, and \( \cos \theta = 0 \).

By understanding this geometric interpretation, we can gain insight into the alignment and relative orientation of vectors in space.

Vector Norms

The vector norm, often referred to as the magnitude or length of a vector, tells us about the size or extent of the vector in its space. For a vector \( \mathbf{v} = (v_1, v_2, v_3) \), the norm is calculated using:
\[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2} \]This computes the Euclidean distance from the origin to the point in space described by the vector:

• A longer vector means a larger norm, while a vector pointing nowhere (like the zero vector) has a norm of zero.
• Norms are always non-negative, reflecting the positive extent or size.
• They are pivotal in establishing vector equality and comparability in terms of scale.

Understanding vector norms is crucial because it lays the foundation for many vector concepts including unit vectors and their roles in directionality.

Parallel Vectors

Two vectors are considered parallel if they have the same or exactly opposite direction. For vectors \( \mathbf{u} \) and \( \mathbf{v} \), they are parallel if there exists a scalar \( k \) such that \( \mathbf{u} = k\mathbf{v} \). In terms of angles:

• If \( \cos \theta = 1 \), the vectors point in the same direction and are parallel.
• If \( \cos \theta = -1 \), the vectors point in opposite directions and are still considered parallel.
• Parallel vectors simplify many vector operations because they effectively represent linear scaling along the same line.

Understanding when vectors are parallel is essential for simplifying problems in physics, engineering, and computer graphics, as it reveals dependencies and shared directions in vector-related scenarios.