Explanation Calculation: Consider the equation, 2 x 2 + 2 y 2 + 2 z 2 + 2 x y + 2 x z + 2 y z − 1 = 0... ( 1 ) The general form of a quadratic equation is, a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + h y + i z + j = 0 Compare Equation (1) with general form, a = 2 , b = 2 , c = 2 , d = 2 , e = 2 , f = 2 , g = 0 , h = 0 , i = 0 and j = − 1 The matrix for general quadratic equation is, X T A X + [ g h i ] X + j = 0... ( 2 ) Here, A = [ a d 2 e 2 d 2 b f 2 e 2 f 2 c ] ... ( 3 ) Substitute 2 for a , 2 for b , 2 for c , 2 for d , 2 for e and 2 for f in Equation (3). A = [ 2 1 1 1 2 1 1 1 2 ] The matrix has Eigenvalues 1 , 1 , 4 , with corresponding eigenvector X 1 = ( 1 , 1 , − 2 ) , X 2 = ( − 1 , 1 , 0 ) and X 3 = ( 1 , 1 , 1 ) . The normalize eigenvector is, u 1 = 1 ( 1 ) 2 + ( 1 ) 2 + ( − 2 ) 2 X 1 ... ( 4 ) Substitute [ 1 1 − 2 ] for X 1 in Equation (4). u 1 = 1 ( 1 ) 2 + ( 1 ) 2 + ( − 2 ) 2 [ 1 1 − 2 ] = 1 6 [ 1 1 − 2 ] = [ 1 6 1 6 − 2 6 ] The normalize eigenvector is, u 2 = 1 ( − 1 ) 2 + ( 1 ) 2 + ( 0 ) 2 X 2 ... ( 5 ) Substitute [ − 1 1 0 ] for X 2 in Equation (5). u 2 = 1 ( − 1 ) 2 + ( 1 ) 2 + ( 0 ) 2 [ − 1 1 0 ] = 1 2 [ − 1 1 0 ] = [ − 1 2 1 2 0 ] The normalize eigenvector is, u 3 = 1 ( 1 ) 2 + ( 1 ) 2 + ( 1 ) 2 X 3 ... ( 6 ) Substitute [ 1 1 1 ] for X 3 in Equation (6). u 3 = 1 ( 1 ) 2 + ( 1 ) 2 + ( 1 ) 2 [ 1 1 1 ] = 1 3 [ 1 1 1 ] = [ 1 3 1 3 1 3 ] Substitute P X ′ for X in Equation (2)

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + MindTap Math, 1 term (6 months) Printed Access Card

8th Edition

ISBN: 9781337131216

Author: Ron Larson

Publisher: Cengage Learning

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Chapter 7.4, Problem 54E

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The matrix A associated with the equation 2x2+2y2+2z2+2xy+2xz+2yz−1=0 and equation of the quadric surface in the rotated x′,y′,z′ system.

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Chapter 7.4, Problem 53E

Chapter 7.4, Problem 55E

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Chapter 7 Solutions

Bundle: Elementary Linear Algebra, Loose-leaf Version, 8th + MindTap Math, 1 term (6 months) Printed Access Card

Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and Eigenvectors in...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Verifying Eigenvalues and EigenvectorsIn Exercises...Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...

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The matrix A associated with the equation 2 x 2 + 2 y 2 + 2 z 2 + 2 x y + 2 x z + 2 y z − 1 = 0 and equation of the quadric surface in the rotated x ′ , y ′ , z ′ system.

Solution Summary: The author analyzes the matrix A associated with the equation, 2x2+2 y2, +2xz+ 2yz-1=0 and equation of the quadric surface in the

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To find:

The matrix A associated with the equation 2x2+2y2+2z2+2xy+2xz+2yz−1=0 and equation of the quadric surface in the rotated x′,y′,z′ system.

Want to see the full answer?

Chapter 7 Solutions

The matrix A associated with the equation 2 x 2 + 2 y 2 + 2 z 2 + 2 x y + 2 x z + 2 y z − 1 = 0 and equation of the quadric surface in the rotated x ′ , y ′ , z ′ system.

Solution Summary: The author analyzes the matrix A associated with the equation, 2x2+2 y2, +2xz+ 2yz-1=0 and equation of the quadric surface in the

Videos

To find: The matrix A associated with the equation 2x2+2y2+2z2+2xy+2xz+2yz−1=0 and equation of the quadric surface in the rotated x′,y′,z′ system.

Want to see the full answer?

Chapter 7 Solutions

To find:

The matrix A associated with the equation 2x2+2y2+2z2+2xy+2xz+2yz−1=0 and equation of the quadric surface in the rotated x′,y′,z′ system.