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If \( L_{i}, m_{i}, n_{i} ; i=1,2,3 \) denote the direction cosines of three mutually perpendicular vectors in space, prove that \( A A^{T}=I \), where \( A=\left[\begin{array}{lll}l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3}\end{array}\right] \).
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