/*! 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 45 Prove that \(\mathbf{u} \times \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 45

Prove that \(\mathbf{u} \times \mathbf{v}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).

Short Answer

Expert verified
By using properties of the dot product and the cross product, we have shown that \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = 0\) and \(\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v}) = 0\), proving that the cross product of \(\mathbf{u}\) and \(\mathbf{v}\) is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\).

Step by step solution

01

Reminder on Dot Product and Orthogonality

We should remember that the dot product of two vectors will be zero if the vectors are orthogonal. Meaning that if vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal, then \(\mathbf{a} \cdot \mathbf{b} = 0\). This is because the dot product is defined as \(\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}| \cos \theta\), where \(\theta\) is the angle between the vectors. If the vectors are orthogonal, \(\theta=90 ^\circ\), and \(\cos(90^\circ) = 0\), thus \(\mathbf{a} \cdot \mathbf{b} = 0\). We'll be using this fact in our proof.
02

Find Dot Product of \(\mathbf{u}\) and \(\mathbf{u} \times \mathbf{v}\)

We need to find the dot product of the vector \(\mathbf{u}\) and the cross product of \(\mathbf{u}\) and \(\mathbf{v}\). By definition, the dot product of these two vectors should be \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v})\). The vector triple product identity states that \(\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) = \mathbf{b} \cdot (\mathbf{c}\times \mathbf{a})\), so substituting \(\mathbf{a} = \mathbf{b} = \mathbf{u}\) and \(\mathbf{c} = \mathbf{v}\), this can be simplified to \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v}) = \mathbf{u} \cdot (\mathbf{v} \times \mathbf{u})\).
03

Use the Anti-Commutative Property of Cross Product

Cross product is anti-commutative, meaning \(\mathbf{b} \times \mathbf{a} = -\mathbf{a} \times \mathbf{b}\). Applying this property to \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{u})\), it becomes \(\mathbf{u} \cdot -(\mathbf{u} \times \mathbf{v})\), which means \(\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v})\) is equal to \(-\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v})\), hence it will be 0.
04

Apply Similar Steps for Vector \(\mathbf{v}\)

We should now repeat this process with \(\mathbf{v}\). Finding the dot product of \(\mathbf{v}\) and the cross product of \(\mathbf{u}\) and \(\mathbf{v}\), we have \(\mathbf{v} \cdot (\mathbf{u} \times \mathbf{v})\). Applying similar logic, we will arrive at the fact that this product also equals 0.

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!

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 the area of the parallelogram that has the vectors as adjacent sides. $$\mathbf{u}=\mathbf{j}, \quad \mathbf{v}=\mathbf{j}+\mathbf{k}$$

Find \(\mathbf{u} \times \mathbf{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v}\). $$\mathbf{u}=\mathbf{i}-2 \mathbf{j}+\mathbf{k}, \quad \mathbf{v}=-\mathbf{i}+3 \mathbf{j}-2 \mathbf{k}$$

Find \(\mathbf{u} \cdot(\mathbf{v} \times \mathbf{w}) .\) This quantity is called the triple scalar product of \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\). $$\mathbf{u}=(1,1,1), \quad \mathbf{v}=(2,1,0), \quad \mathbf{w}=(0,0,1)$$

The table shows the sales \(y\) (in millions of dollars) for Dell Incorporated during the years 1996 to 2007 . Find the least squares regression line and the least squares cubic regression polynomial for the data. Let \(t\) represent the year, with \(t=-4\) corresponding to \(1996 .\) Which model is the better fit for the data? Why? (Source: Dell Inc.) $$\begin{aligned} &\begin{array}{l|llll} \hline \text {Year} & 1996 & 1997 & 1998 & 1999 \\ \text {Sales, } y & 7759 & 12,327 & 18,243 & 25,265 \\ \hline \text {Year} & 2000 & 2001 & 2002 & 2003 \\ \text {Sales, } y & 31,888 & 31,168 & 35,404 & 41,444 \\ \hline \end{array}\\\ &\begin{array}{l|llll} \hline \\ \hline \text {Year} & 2004 & 2005 & 2006 & 2007 \\ \text {Sales, } y & 49,205 & 55,908 & 58,200 & 61,000 \\ \hline \end{array} \end{aligned}$$

(a) Prove that \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \cdot \mathbf{w}) \mathbf{v}-(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}\). (b) Find an example for which \(\mathbf{u} \times(\mathbf{v} \times \mathbf{w}) \neq(\mathbf{u} \times \mathbf{v}) \times \mathbf{w}\).
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

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.