How to know which surface represents equation Q (x,y,z) =0?

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The discussion focuses on the equation Q(x, y, z) = -5/2x² - y² + 4z² + 7xy - 2xz - 2yz, which represents a surface in three-dimensional space. To analyze this surface, one must derive its axis and determine its intersection with the plane defined by x + y + z = 0. The equation can be expressed using a symmetric matrix, which facilitates the calculation of eigenvalues and eigenvectors, enabling a transformation to a coordinate system devoid of mixed terms.

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The equation.
Q(x, y,z) = -5/2:X2 - y2 + 4z2 + 7xy - 2xz - 2yz.

Find its axis and draw its intersection with the plane x + y + z = 0 .
 
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Q(x,y,z) can be written as
[tex]\begin{bmatrix}x & y & z \end{bmatrix}\begin{bmatrix}1 & \frac{7}{2} & -1 \\ \frac{7}{2} & -1 & -1 \\ -1 & -1 & 4 \end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}[/tex].
Where I have divided the coefficients of xy, xz, and yz equally to get a symmetric matrix Symmetric matrices always have a "complete set" of independent eigenvectors.

Finding the eigenvalues and eigenvectors of that matrix will allow you to write the equation in a new coordinate system that has no "mixed" terms.
 
Misplaced homework question, so I am locking the thread.
 

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