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

[starterYEAR]

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 .

 

[HallsofIvy]

Q(x,y,z) can be written as
\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}.
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.

 

[Mark44]

Misplaced homework question, so I am locking the thread.