/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 a. Cauchy-Schwartz inequality si... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 27

a. Cauchy-Schwartz inequality since \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) 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.

Short Answer

Expert verified
The inequality holds always; equality occurs when vectors are parallel or antiparallel.

Step by step solution

01

Understand Cauchy-Schwarz Inequality

The Cauchy-Schwarz inequality states that for any vectors \( \mathbf{u} \) and \( \mathbf{v} \), the dot product satisfies:\[ |\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}| \]This means the absolute value of the dot product of two vectors is less than or equal to the product of their magnitudes.
02

Express the Dot Product in Terms of Angle

The dot product \( \mathbf{u} \cdot \mathbf{v} \) can be expressed as:\[ \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \cos \theta \]where \( \theta \) is the angle between the two vectors. This expression helps in analyzing the behavior of the dot product based on the angle.
03

Analyze the Expression for Maximum and Minimum Values

Since \( \cos \theta \) can only range between -1 and 1, the dot product becomes:\[ -|\mathbf{u}||\mathbf{v}| \leq \mathbf{u} \cdot \mathbf{v} \leq |\mathbf{u}||\mathbf{v}| \]Thus, the absolute value of the dot product will always be less than or equal to the product of the magnitudes, that is:\[ |\mathbf{u} \cdot \mathbf{v}| \leq |\mathbf{u}||\mathbf{v}| \]
04

Determine When the Equality Holds

The equality \( |\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}| \) holds when \( \cos \theta = \pm 1 \). This occurs when:- \( \theta = 0 \) degrees (vectors are parallel or in the same direction) or- \( \theta = 180 \) degrees (vectors are antiparallel or oppositely directed).In these cases, the cosine function achieves its maximum or minimum value, making the absolute value of the dot product equal to the product of the magnitudes.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dot Product
The dot product, also known as the scalar product, is a mathematical operation that takes two vectors and returns a single scalar value. It is calculated by multiplying the corresponding components of two vectors and then adding these up.
In mathematical terms, if we have two vectors, \( \mathbf{u} = [u_1, u_2, ..., u_n] \) and \( \mathbf{v} = [v_1, v_2, ..., v_n] \), their dot product is given by:\[ \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + ... + u_nv_n \]
  • The result is a scalar, not a vector.
  • It's an important tool in physics and engineering as it can determine how much one vector goes in the direction of another.
This result is not just a mathematical curiosity but provides a deep insight into the properties of geometry and helps evaluate angles between vectors.
Vector Magnitude
The magnitude of a vector, often denoted as \(|\mathbf{u}|\) for a vector \(\mathbf{u}\), is a measure of its length. You can think of it like the 3D version of the Pythagorean theorem.
If we have a vector \(\mathbf{u} = [u_x, u_y, u_z]\), its magnitude is calculated as:\[ |\mathbf{u}| = \sqrt{u_x^2 + u_y^2 + u_z^2} \]
  • The magnitude is always a non-negative number.
  • It tells us the size or length of the vector.
Understanding vector magnitude is crucial for applying the Cauchy-Schwarz inequality, especially when relating the dot product to vector magnitudes.
Angle Between Vectors
The angle between vectors is a measure of how one vector is oriented in relation to another. The direction and orientation can be crucial when comparing two vectors.
To find the angle \(\theta\) between two vectors \(\mathbf{u}\) and \(\mathbf{v}\), we use the formula involving the dot product:\[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}||\mathbf{v}|} \]
  • This equation derives from rearranging the dot product relation to isolate \(\cos \theta\).
  • The angle can tell us if the vectors are moving towards the same point, diverging, or moving in opposite directions.
Knowing the angle helps to determine when the equality condition of the Cauchy-Schwarz inequality holds, as it requires \(\cos \theta = \pm 1\).
Parallel Vectors
Vectors are considered parallel when they have the same or opposite direction. This occurs when the angle between them is 0 degrees, meaning they line up perfectly, or 180 degrees, meaning they are pointing in precisely opposite directions.
When vectors are parallel, we have:\( \cos \theta = 1 \) for the same direction, and \( \cos \theta = -1 \) for opposite direction.
  • If \(\mathbf{u}\) and \(\mathbf{v}\) are non-zero vectors with \(\theta = 0\) degrees, they are parallel in the same direction.
  • Their dot product equals the product of their magnitudes: \( \mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}| \).
Parallel vectors help in maximizing or minimizing the value of \(\cos \theta\), which makes the dot product equal to the product of magnitudes.
Antiparallel Vectors
Antiparallel vectors are vectors that have the same magnitude but are oriented in opposite directions. This is mathematically described when the angle between two vectors is 180 degrees.
In the case of antiparallel vectors:
  • The cosine of the angle \( \theta = 180 \) degrees results in \( \cos \theta = -1 \), making the vectors point exactly opposite directions.
  • Even though their direction is opposite, the concept of magnitude remains unchanged.
  • The dot product becomes negative but its absolute value is the product of their magnitudes: \(|\mathbf{u} \cdot \mathbf{v}| = |\mathbf{u}||\mathbf{v}|\).
Understanding antiparallel vectors goes hand-in-hand with parallel vectors, as both are essential in fully grasping the Cauchy-Schwarz inequality and when it achieves equality.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find equations for the spheres whose centers and radii are given in Exercises \(61-64\) . $$\frac{\text { Center }}{(0,-1,5)} \frac{\text { Radius }}{2}$$

\begin{equation} \begin{array}{l}{\text { In Exercises } 35-44 \text { , describe the given set with a single equation or }} \\ {\text { with a pair of equations. }}\end{array} \end{equation} The circle of radius 2 centered at \((0,0,0)\) and lying in the \begin{equation} \text { a. }x y \text { -plane } \quad \text { b. } y z \text { -plane } \quad \text { c. } x z \text { -plane } \end{equation}

Find the point equidistant from the points \((0,0,0),(0,4,0),(3,0,0),(3,0,0),\) and \((2,2,-3) .\)

Find the distance from the \begin{equation} \begin{array}{l}{\text { a. } x \text { -axis to the plane } z=3 .} \\ {\text { b. origin to the plane } 2=z-x \text { . }} \\ {\text { c. point }(0,4,0) \text { to the plane } y=x \text { . }}\end{array} \end{equation}

Find equations for the spheres whose centers and radii are given. $$ \begin{array}{lc} \text { Center } & \text { Radius } \\ \hline(1,2,3) & \sqrt{14} \end{array} $$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.