Explanation Calculation: Consider the equation, 3 x 2 − 2 x y + 3 y 2 + 8 z 2 − 16 = 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 = 3 , b = 3 , c = 8 , d = − 2 , e = 0 , f = 0 , g = 0 , h = 0 , i = 0 and j = − 16 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 3 for a , 3 for b , 8 for c , − 2 for d , 0 for e and 0 for f in Equation (3). A = [ 3 − 1 0 − 1 3 0 0 0 8 ] The matrix has Eigenvalues 2 , 4 , 8 , with corresponding eigenvector X 1 = ( 1 , 1 , 0 ) , X 2 = ( − 1 , 1 , 0 ) and X 3 = ( 0 , 0 , 1 ) . The normalize eigenvector is, u 1 = 1 ( 1 ) 2 + ( 1 ) 2 + ( 0 ) 2 X 1 ... ( 4 ) Substitute [ 1 1 0 ] for X 1 in Equation (4). u 1 = 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 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 ( 0 ) 2 + ( 0 ) 2 + ( 1 ) 2 X 3 ... ( 6 ) Substitute [ 0 0 1 ] for X 3 in Equation (6). u 3 = 1 ( 0 ) 2 + ( 0 ) 2 + ( 1 ) 2 [ 0 0 1 ] = [ 0 0 1 ] Substitute P X ′ for X in Equation (2)

Elementary Linear Algebra - Text Only (Looseleaf)

8th Edition

ISBN: 9781305953208

Author: Larson

Publisher: Cengage Learning

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Chapter 7.4, Problem 53E

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The matrix A associated with the equation 3x2−2xy+3y2+8z2−16=0 and equation of the quadric surface in the rotated x′,y′,z′ system.

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Chapter 7.4, Problem 52E

Chapter 7.4, Problem 54E

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Elementary Linear Algebra - Text Only (Looseleaf)

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 3 x 2 − 2 x y + 3 y 2 + 8 z 2 − 16 = 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 3x2-2xy+3y2,+8z 2-16=0 and equation of the quadric surface in the

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To find:

The matrix A associated with the equation 3x2−2xy+3y2+8z2−16=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 3 x 2 − 2 x y + 3 y 2 + 8 z 2 − 16 = 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 3x2-2xy+3y2,+8z 2-16=0 and equation of the quadric surface in the

Videos

To find: The matrix A associated with the equation 3x2−2xy+3y2+8z2−16=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 3x2−2xy+3y2+8z2−16=0 and equation of the quadric surface in the rotated x′,y′,z′ system.