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Engineering Calculus Notes 387

Engineering Calculus Notes 387 - 3.9 THE PRINCIPAL AXIS...

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3.9. THE PRINCIPAL AXIS THEOREM 375 What does this result tell us about quadratic forms? We start with some consequences of the second property of the eigenvectors −→ u i : −→ u i · −→ u j = braceleftBigg 0 if i negationslash = j, 1 if i = j. Geometrically, this says two things: first (using i = j ) they are unit vectors ( | −→ u i | 2 −→ u i · −→ u i = 1), and second, they are mutually perpendicular. A collection of vectors with both properties is called an orthonormal set of vectors. Since they define three mutually perpendicular directions in space, a set of three orthonormal vectors in R 3 can be used to set up a new rectangular coordinate system: any vector −→ x R 3 can be located by its projections onto the directions of these vectors, which are given as the dot products of −→ x with each of the −→ u i . You should check that using these coordinates, we can express any vector −→ x R 3 as a linear combination of the −→ u i : −→ x = ( −→ x · −→ u 1 ) −→ u 1 + ( −→ x ·
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