/*! 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 44 Prove the property of the cross ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 44

Prove the property of the cross product. $$ \mathbf{u} \times \mathbf{u}=\mathbf{0} $$

Short Answer

Expert verified
The cross product of a vector by itself results in the zero vector. This is due to the fact that every cross multiplication of identical components will result in 0, giving a final cross product of 0.

Step by step solution

01

Understanding Cross Product

Cross product, also known as vector product, gives a vector which is perpendicular to the vectors being multiplied. The result vector has a direction according to the right-hand rule and a magnitude equal to the area of the parallelogram that the vectors span.
02

Express the vectors

Since \(\mathbf{u}\) is a single vector, it's feasible to label it's components as \(u_i\), \(u_j\), and \(u_k\) for the x, y, and z dimensions respectively. Therefore, \(\mathbf{u} = u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\). The cross product is then, \(\mathbf{u} \times \(\mathbf{u}\), which is \(u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\) cross \(u_i\hat{i} + u_j\hat{j} + u_k\hat{k}\).
03

Compute the cross product

Cross product is normally computed using the determinant of a special 3x3 matrix, but it can also be computed component by component. According to the properties of the cross products, \(i \times i = j \times j = k \times k = 0\). Therefore, every cross multiplication of identical components will result in 0, and as a result, the entire cross product will be 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

The vertices of a triangle are given. Determine whether the triangle is an acute triangle, an obtuse triangle, or a right triangle. Explain your reasoning. $$ (-3,0,0),(0,0,0),(1,2,3) $$

Find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\mathbf{i}, \quad \mathbf{v}=\mathbf{i} $$

Find the angle \(\theta\) between the vectors. $$ \mathbf{u}=\cos \left(\frac{\pi}{6}\right) \mathbf{i}+\sin \left(\frac{\pi}{6}\right) \mathbf{j}, \quad \mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j} $$

Write an equation whose graph consists of the set of points \(P(x, y, z)\) that are twice as far from \(A(0,-1,1)\) as from \(B(1,2,0)\)

Find the vector \(z,\) given that \(\mathbf{u}=\langle 1,2,3\rangle\) \(\mathbf{v}=\langle 2,2,-1\rangle,\) and \(\mathbf{w}=\langle 4,0,-4\rangle\) \(\mathbf{z}=5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} \mathbf{w}\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics 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.