JEE Advance - Mathematics (2004 - No. 2)
$${P_1}$$ and $${P_2}$$ are planes passing through origin. $${L_1}$$ and $${L_2}$$ are two line on $${P_1}$$ and $${P_2}$$ respectively such that their intersection is origin. Show that there exists points $$A, B, C,$$ whose permutation $$A',B',C'$$ can be chosen such that (i) $$A$$ is on $${L_1},$$ $$B$$ on $${P_1}$$ but not on $${L_1}$$ and $$C$$ not on $${P_1}$$ (ii) $$A'$$ is on $${L_2},$$ $$B'$$ on $${P_2}$$ but not on $${L_2}$$ and $$C'$$ not on $${P_2}$$
The statement is always true and can be proved using linear independence and suitable vector choices.
The statement is false in general because the conditions impose strong constraints on the relative orientations of the planes and lines.
The existence of such points depends on the angle between the planes $${P_1}$$ and $${P_2}$$.
If $${L_1}$$ and $${L_2}$$ are orthogonal, then such points always exist.
The problem can be solved by finding the matrix transformation which maps $${L_1}$$ to $${L_2}$$ and verifying existence.
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