19. If li,mi,ni; i=1,2,3 denote the direction cosines of three mutually perpendicular vectors in space,prove that AA'=I,where A=[l1 m1 n1 ] [ l2 m2 n2] [l3 m3 n3 ]

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If $l_{1}, m_{1}, n_{1}, l_{2}, m_{2}, n_{2} a n d l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,then prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3} a n d n_{1} + n_{2} + n_{3}$ makes equal angles with them.

Q. lf

(l_{1}, m_{1}, n_{1})

(l_{2}, m_{2}, n_{2})

(l_{3}, m_{3}, n_{3})

are the d.c.s of three mutually perpendicular lines, then the direction cosines of the line whose direction ratios are

l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}, n_{1} + n_{2} + n_{3}

are

Q. If l_i, m_i, n_i; i = 1, 2, 3 denote the direction cosines of three mutually perpendicular vectors in space, prove that AA^T = I, where

A = [\begin{matrix} l_{1} & m_{1} & n_{1} \\ l_{2} & m_{2} & n_{2} \\ l_{3} & m_{3} & n_{3} \end{matrix}]

If three mutually perpendicular lines have direction cosines $(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})$ and $(l_{3}, m_{3}, n_{3})$ line having direction cosines $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}$ and $n_{1} + n_{2} + n_{3}$ make an
angle of..... with each other