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Question

# Determine a rotation $\mathbf{x}=P \mathbf{x}^{\prime}$ that removes the cross-product terms, and give its equation in the $x^{\prime} y^{\prime} z^{\prime}$-system. Name the quadric.$2 x^2+3 y^2+23 z^2+72 x z+150=0$

Solution

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Step 1
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Writing the quadric's equation in matrix form:

${\bf x}^{T} A {\bf x}+j=0 ,$

where,

${\bf x}=\left[\begin{array}{c} x \\y \\z \end{array}\right]\;, \quad A=\left[\begin{array}{ccc} 2 & 0 & 36 \\ 0 & 3 & 0 \\ 36 & 0 & 23 \end{array}\right]\;, \quad j=150$

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