OA, OB and OC, with O as the origin, are three mutually perpendicular lines whose lines whose direction cosines are `l_(r),m_(r)andn_(r)(r=1,2 and 3)`. If the projections of OA and OB on the plane z=0 make angles `phi_(1)andphi_(2)`, respectively, with the x-axis, prove that `tan(phi_(1)-phi_(2))=+-n_(3)//n_(1)n_(2)`.

Explanation

To solve the problem, we need to prove that:

\[ \tan(\phi_1 - \phi_2) = \pm \frac{n_3}{n_1 n_2} \]

where \( \phi_1 \) and \( \phi_2 \) are the angles made by the projections of lines OA and OB on the plane \( z = 0 \) with the x-axis, and \( n_1, n_2, n_3 \) are the direction cosines of the lines OA, OB, and OC respectively.

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