8.67 (a), §2 Solution

2 x² - 6 xy + 10 y²

Find the matrix of coefficients:

Eigenvalues:

There are two roots: and

The eigenvector corresponding to satisfies

Taking v_1y = 1, then v_1x = 3, giving an eigenvector (3,1). Normalizing this vector to length one gives:

The eigenvector corresponding to satisfies

Taking v_2y = 3, then v_2x =-1, giving after normalization:

Since , the new axes are rotated counter clockwise from the old:

In the new coordinates,

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