Miscellaneous

Chapter 11 Class 12 Three Dimensional Geometry
Serial order wise

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Misc 2 If π_1 , π_1, π_1 and π_2 , π_2, π_2 are the direction cosines of two mutually perpendicular lines, show that direction cosines of line perpendicular to both of these are π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1.We know that π β Γ π β is perpendicular to both π β & π β So, required line is cross product of lines having direction cosines π_1 , π_1, π_1 and π_2 , π_2, π_2 Required line = |β 8(π Μ&π Μ&π Μ@π_1&π_1&π[email protected]π_2&π_2&π_2 )| = π Μ (π_1 π_2 β π_2 π_1) β π Μ (π_1 π_2 β π_2 π_1) + π Μ(π_1 π_2 β π_2 π_1) = (π_1 π_2 β π_2 π_1) π Μ + (π_2 π_1βπ_1 π_2) π Μ + (π_1 π_2 β π_2 π_1) π Μ Hence, direction cosines = π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1 β΄ Direction cosines of the line perpendicular to both of these are π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1 , π_1 π_2 β π_2 π_1. Hence proved