If `l_(1), m_(1), n_(1)` and `l_(2),m_(2),n_(2)` are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are `m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1)`.A. `m_(1)n_(2)+m_(2)n_(1).n_(1)l_(2)+n_(2)l_(1).l_(1)m_(2)+l_(2)m_(1)`B. `m_(1)n_(2)-m_(2)n_(1).n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1)`C. `m_(1)m_(2)-n_(1)n_(2).n_(1)n_(2)-l_(1)l_2,l_(1)l_(2)-m_(1)m_(2)`D. `m_(1)m_(2)+n_(1)n_(2).n_(1)n_(2)-l_(1)l_2,l_(1)l_(2)-m_(1)m_(2)`

If `l_(1), m_(1), n_(1)` and `l_(2),m_(2),n_(2)` are the direction cos

1.	If `l_(1), m_(1), n_(1)` and `l_(2),m_(2),n_(2)` are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are `m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1)`.A. `m_(1)n_(2)+m_(2)n_(1).n_(1)l_(2)+n_(2)l_(1).l_(1)m_(2)+l_(2)m_(1)`B. `m_(1)n_(2)-m_(2)n_(1).n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1)`C. `m_(1)m_(2)-n_(1)n_(2).n_(1)n_(2)-l_(1)l_2,l_(1)l_(2)-m_(1)m_(2)`D. `m_(1)m_(2)+n_(1)n_(2).n_(1)n_(2)-l_(1)l_2,l_(1)l_(2)-m_(1)m_(2)`
Answer» Correct Answer - B Given lines are represented parallel to unit vectors `b_1=i_(1)hati+m_(1)hatj+n_(1)hatk....(i)` `and b_2=i_(2)hati+m_(2)hatj+n_(2)hatk....(ii)` Now, `b_(1) xx b_(2)` is a vector which is at right angles to both `b_1 and b_2` and is of magnitude unity. Henec, components of `b_(1) xx b_(2)` are directions cosines of a line which is at right angle to both `b_1 and b_2` `=(m_1n_2-m_2n_1)hati+(n_(1)l_(2)-n_(2)l_(1))hatj+(l_(1)m_(2)-l_(2)m_(1))hatk` Thus, the direction cosiness of the required line are `m_(1)n_(2)-m_(2)n_(1), n_(1)l_(2)-n_(2)l_(1),m_(2)-l_(2)m_(1)`

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