/*! 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 the property of the cross ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 45

Prove the property of the cross product. $$ \mathbf{u} \cdot(\mathbf{v} \times \mathbf{w})=(\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} $$

Short Answer

Expert verified
The scalar triple product identity \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = (\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w}\) has been proven by expressing the vectors in component form, calculating the cross products, substituting back in, and showing that the results of the two sides of the identity are equal.

Step by step solution

01

Apply the coordinates

Consider vectors \(\mathbf{u}=[u_1,u_2,u_3]\), \(\mathbf{v}=[v_1,v_2,v_3]\) and \(\mathbf{w}=[w_1,w_2,w_3]\). Substitute these into the equation to specify the problem in terms of identifiable components.
02

Calculate cross products

Calculate the cross product \(\mathbf{v} \times \mathbf{w}\) and \(\mathbf{u} \times \mathbf{v}\) using the rule of cross product in 3D. \(\mathbf{v} \times \mathbf{w} = [v_2w_3 - v_3w_2, v_3w_1 - v_1w_3, v_1w_2 - v_2w_1]\) and \(\mathbf{u} \times \mathbf{v} = [u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1]\).
03

Substitute cross products back into the identity

Substitute the calculated cross products back into the identity, and calculate the dot product for each side. \(\mathbf{u} \cdot (\mathbf{v} \times \mathbf{w}) = u_1(v_2w_3 - v_3w_2) + u_2(v_3w_1 - v_1w_3) + u_3(v_1w_2 - v_2w_1)\), and \((\mathbf{u} \times \mathbf{v}) \cdot \mathbf{w} = w_1(u_2v_3 - u_3v_2) + w_2(u_3v_1 - u_1v_3) + w_3(u_1v_2 - u_2v_1)\). After calculations, you should see they result in the same expression, thereby proving the identity.

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 vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=2 \mathbf{u}+4 \mathbf{v}-\mathbf{w}\)

In Exercises \(65-68,\) find the magnitude of \(v\). \(\mathbf{v}=\mathbf{i}-2 \mathbf{j}-3 \mathbf{k}\)

The vector \(\mathbf{u}=\langle 3240,1450,2235\rangle\) gives the numbers of hamburgers, chicken sandwiches, and cheeseburgers, respectively, sold at a fast-food restaurant in one week. The vector \(\mathbf{v}=\langle 1.35,2.65,1.85\rangle\) gives the prices (in dollars) per unit for the three food items. Find the dot product \(\mathbf{u} \cdot \mathbf{v},\) and explain what information it gives.

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=\mathbf{u}-\mathbf{v}+2 \mathbf{w}\)

In Exercises 69 and \(70,\) find a unit vector \((a)\) in the direction of \(\mathbf{u}\) and \((\mathbf{b})\) in the direction opposite \(\mathbf{u}\) \(\mathbf{u}=\langle 2,-1,2\rangle\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation

Discrete 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.