Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Matrix Algebra Rate of Change Rate of temperature change, degrees/hour -5 Chapter 10 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour Applied Force F = 200 lbf 20 feet 100 200 300 400 θ = 600 500 100 200 300 400 500 Pivot Point MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. 10.1 Matrix Operations and Functions Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 • Matrix algebra is used extensively in engineering applications • Matrix algebra is different from the array calculations we have performed thus far 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Array Operators Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 • A.* B time, hour • A./B 100 200 300 400 500 100 200 300 400 500 • A.^B multiplies each element in array A times the corresponding element in array B divides each element in array A by the corresponding element in array B raises each element in array A to the power in the corresponding element of array B MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Operators used in Matrix Mathematics Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 • • • • • Transpose Multiplication Division Exponentiation Left Division 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Some Matrix Algebra functions Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 • • • • Dot products Cross products Inverse Determinants 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Transpose Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • In mathematics texts you will often see the transpose indicated with superscript T • AT 100 200 300 400 500 100 200 300 400 500 • The MATLAB syntax for the transpose is • A' MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The transpose switches the rows and columns Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 1 2 3 4 5 6 A 7 8 9 10 11 12 100 200 300 1 4 7 10 AT 2 5 8 11 3 6 9 12 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Using the transpose with complex numbers Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 When used with complex numbers, the transpose operator returns the complex conjugate MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Dot Products Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 • The dot product is sometimes called the scalar product • the sum of the results when you multiply two vectors together, element by element. 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 * * * || || -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour || + + 100 Equivalent statements 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Example 10.1 Calculating the Center of Gravity Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 • Finding the center of gravity of a structure is important in a number of engineering applications • The location of the center of gravity can be calculated by dividing the system up into small components. MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. xW x1W1 x2W2 x3W3 etc... yW y1W1 y2W2 y3W3 etc... Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour zW z1W1 z2W2 z3W3 etc... • In a rectangular coordinate system • x , y , and z are the coordinates of the center of gravity 100 200 300 400 500 100 200 300 400 500 • W is the total mass of the system • x1, x2, and x3 etc are the x coordinates of each system component • y1, y2, and y3 etc are the y coordinates of each system component • z1, z2, and z3 etc are the z coordinates of each system component and • W1, W2, and W3 etc are the weights of each system component MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. In this example… Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 • We’ll find the center of gravity of a small collection of the components used in a complex space vehicle 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Vehicle Component Locations and Mass Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour Item x, meters y, meters z meters Mass Bolt 0.1 2 3 3.50 gram screw 1 1 1 1.50 gram nut 1.5 0.2 0.5 0.79 gram bracket 2 2 4 1.75 gram 100 200 300 400 500 100 200 300 400 500 Formulate the problem using a dot product MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. We could use a plot to evaluate our results Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 Center of Gravity -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 4 z-axis 3 This plot was enhanced using the interactive plotting tools Center of Gravity 2 100 1 200 300 0 2 400 2 500 100 200 300 400 500 1 y-axis 1 0 0 x-axis MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Matrix Multiplication Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • Similar to a dot product 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 • Matrix multiplication results in an array where each element is a dot product. • In general, the results are found by taking the dot product of each row in matrix A with each column in Matrix B MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour N Ci , j Ai ,k Bk , j k 1 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 We could use matrix multiplication to solve the problem in Example 10.1, in a single step MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Matrix Powers Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • Raising a matrix to a power is equivalent to multiplying it times itself the requisite number of times • A2 is the same as A*A • A3 is the same as A*A*A 100 200 300 400 500 100 200 300 400 500 • Raising a matrix to a power requires it to have the name number of rows and columns MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Matrix Inverse Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • MATLAB offers two approaches • The matrix inverse function • inv(A) • Raising a matrix to the -1 power 100 • A-1 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour Equivalent approaches to finding the inverse of a matrix 100 200 300 400 500 100 200 300 400 500 A matrix times its inverse is the identity matrix MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Not all matrices have an inverse Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • Called • Singular • Ill-conditioned matrices 100 200 300 400 500 100 200 300 400 500 • Attempting to take the inverse of a singular matrix results in an error statement MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Determinants Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 • Related to the matrix inverse • If the determinant is equal to 0, the matrix does not have an inverse • The MATLAB function to find a determinant is • det(A) MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Cross Products Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 • sometimes called vector products • the result of a cross product is a vector • always at right angles (normal) to the plane defined by the two input vectors • orthogonality MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Consider two vectors Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 A Ax i Ay j Az k B Bx i B y j Bz k 200 300 The cross product is equal to… 400 500 100 200 300 400 500 A B ( Ay * Bz Az * B y )i ( Az * Bx Ax * Bz ) j ( Ax B y Ay Bx )k MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Cross Products are Widely Used Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 • Cross products find wide use in statics, dynamics, fluid mechanics and electrical engineering problems 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. 10.2 Solutions to Systems of Linear Equations Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 3x 2 y x x 100 200 z 10 3y 2 z 5 y z 1 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Using Matrix Nomenclature Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 3 2 1 x A 1 3 2 X y z 1 1 1 10 B 5 1 100 and 200 300 400 500 100 200 300 400 500 AX=B MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. We can solve this problem using the matrix inverse approach Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 200 300 400 500 100 200 300 400 500 This approach is easy to understand, but its not the more efficient computationally MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour 100 Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. 10.3 Special Matrices Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour • We introduced some of MATLAB’s special matrices in previous chapters • ones • zeros 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The identity matrix is another special matrix that is useful in Matrix Algebra Rate of Change Rate of temperature change, degrees/hour -5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 time, hour It may be tempting to name an identity matrix i, however i is already in-use for imaginary numbers 100 200 300 400 500 100 200 300 400 500 MATLAB for Engineers 3E, by Holly Moore. © 2011 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.