3.9. THE PRINCIPAL AXIS THEOREM375What does this result tell us about quadratic forms?We start with some consequences of the second property of theeigenvectors−→ui:−→ui·−→uj=braceleftBigg0ifinegationslash=j,1ifi=j.Geometrically, this says two things: first (usingi=j) they areunitvectors(|−→ui|2−−→ui·−→ui= 1), and second, they are mutually perpendicular. Acollection of vectors with both properties is called anorthonormalset ofvectors. Since they define three mutually perpendicular directions in space,a set of three orthonormal vectors inR3can be used to set up a newrectangular coordinate system: any vector−→x∈R3can be located by itsprojections onto the directions of these vectors, which are given as the dotproducts of−→xwith each of the−→ui. You should check that using thesecoordinates, we can express any vector−→x∈R3as a linear combination ofthe−→ui:−→x= (−→x·−→u1)−→u1+ (−→x·
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