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In order to proof the orthogonal condition a_{ij}a_{ik}=[tex]\delta_{jk}[/tex] j,k=1,2,3

I write the invariance of the length of a vector in two coordinate systems:

x'_{i}x'_{i}=x_{i}x_{i}

Using the linear transformation:

x'_{i}=a_{i1}x_{i1}+a_{i2}x_{i2}+a_{i3}x_{i3}

the first term becomes:

a_{ij}a_{ik}x_{j}x_{k}

My question is: why can't I write

a_{ij}^{2}=x_{j}^{2}

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# Orthogonal Transformation and condition

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