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3. A quadric surface is a surface in R³ defined by an equation of the form ax² + by² + cz² + dxy + exz + fyz + gx+hy+iz + j = 0. After a suitable rotation and a translation, it is possible to convert the equation into one of the standard forms listed in the table here: https://en.wikipedia.org/wiki/Quadric#Euclidean_space The general method for converting the equation is as follows. First, rewrite the equation in matrix notation as where Then X ₁ (*) + (-) +, B (x y z) Ay (x y z) A A = (ghi). Since A is real and symmetric, by the spectral theorem there exist an orthogonal matrix Q and a diagonal matrix D such that QAQ = D. One may choose Q with a positive determinant, so that it represents a rotation by Euler's rotation theorem. Let a d/2 e/2\ d/2 b ƒ/2 B e/2 f/2 C () -- () Qt X ») ^ ( ) + 0 (*) + > - B + j = 0, = +j= (x y z¹) Dy+B'y' + j 1 (a) 2xy + 2xz+ 2yz - 6x - 4y - 6z+ 9 = 0. 21 where B' BQ. Finally, find a suitable translation by completing the square to put the equation into the standard form. 21 Convert the following two equations into the standard form to determine what types of quadric surfaces they define.

3. A quadric surface is a surface in R³ defined by an equation of the form ax² + by² + cz² + dxy + exz + fyz + gx+hy+iz + j = 0. After a suitable rotation and a translation, it is possible to convert the equation into one of the standard forms listed in the table here: https://en.wikipedia.org/wiki/Quadric#Euclidean_space The general method for converting the equation is as follows. First, rewrite the equation in matrix notation as where Then X ₁ (*) + (-) +, B (x y z) Ay (x y z) A A = (ghi). Since A is real and symmetric, by the spectral theorem there exist an orthogonal matrix Q and a diagonal matrix D such that QAQ = D. One may choose Q with a positive determinant, so that it represents a rotation by Euler's rotation theorem. Let a d/2 e/2\ d/2 b ƒ/2 B e/2 f/2 C () -- () Qt X ») ^ ( ) + 0 (*) + > - B + j = 0, = +j= (x y z¹) Dy+B'y' + j 1 (a) 2xy + 2xz+ 2yz - 6x - 4y - 6z+ 9 = 0. 21 where B' BQ. Finally, find a suitable translation by completing the square to put the equation into the standard form. 21 Convert the following two equations into the standard form to determine what types of quadric surfaces they define.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter11: Topics From Analytic Geometry
Section11.3: Hyperbolas
Problem 36E
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3. A quadric surface is a surface in R³ defined by an equation of the form ax² + by² + cz²+dxy + exz + fyz + gx +hy+iz + j = 0. After a suitable rotation and a translation, it is possible to convert the equation into one of the standard forms listed in the table here: https://en.wikipedia.org/wiki/Quadric#Euclidean_space The general method for converting the equation is as follows. First, rewrite the equation in matrix notation as where (x y z) A Then a d/2 A = d/2 b X 2 e/2 f/2 C + B X e/2 ƒ/2 B=(g_hi). 2 Z Since A is real and symmetric, by the spectral theorem there exist an orthogonal matrix Q and a diagonal matrix D such that QtAQ = D. One may choose Q with a positive determinant, so that it represents a rotation by Euler's rotation theorem. Let + j = 0, 0--0 = Qt 1 X (x y z) A · · * (?) + ² (1) − × ( ) +- B +j= (x' y' z') Dy' + j اج where B' BQ. Finally, find a suitable translation by completing the square to put the equation into the standard form. Convert the following two equations into the standard form to determine what types of quadric surfaces they define. (a) 2xy + 2xz + 2yz - 6x - 4y - 6z+ 9 = 0. (b) 5x² + 2y² + 2z² − 2xy + 2xz − 4yz - 2x+10y - 26z = 0.
Transcribed Image Text:3. A quadric surface is a surface in R³ defined by an equation of the form ax² + by² + cz²+dxy + exz + fyz + gx +hy+iz + j = 0. After a suitable rotation and a translation, it is possible to convert the equation into one of the standard forms listed in the table here: https://en.wikipedia.org/wiki/Quadric#Euclidean_space The general method for converting the equation is as follows. First, rewrite the equation in matrix notation as where (x y z) A Then a d/2 A = d/2 b X 2 e/2 f/2 C + B X e/2 ƒ/2 B=(g_hi). 2 Z Since A is real and symmetric, by the spectral theorem there exist an orthogonal matrix Q and a diagonal matrix D such that QtAQ = D. One may choose Q with a positive determinant, so that it represents a rotation by Euler's rotation theorem. Let + j = 0, 0--0 = Qt 1 X (x y z) A · · * (?) + ² (1) − × ( ) +- B +j= (x' y' z') Dy' + j اج where B' BQ. Finally, find a suitable translation by completing the square to put the equation into the standard form. Convert the following two equations into the standard form to determine what types of quadric surfaces they define. (a) 2xy + 2xz + 2yz - 6x - 4y - 6z+ 9 = 0. (b) 5x² + 2y² + 2z² − 2xy + 2xz − 4yz - 2x+10y - 26z = 0.
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