Question

# 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 .

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Solution

## The direction cosines of two mutually perpendicular lines are given as l 1 , m 1 , n 1 and l 2 , m 2 , n 2 . Since the lines are perpendicular, so, l 1 l 2 + m 1 m 2 + n 1 n 2 =0 l 1 2 + m 1 2 + n 1 2 =1 l 2 2 + m 2 2 + n 2 2 =1 Let the direction cosine of the line that is perpendicular to line with given direction cosines be l,m,n. So, l l 1 +m m 1 +n n 1 =0 And, l l 2 +m m 2 +n n 2 =0 The above equations can be written as, l m 1 n 2 − n 2 n 1 = m n 1 l 2 − n 2 l 1 = n l 1 m 2 − l 2 m 1 ( l m 1 n 2 − n 2 n 1 ) 2 = ( m n 1 l 2 − n 2 l 1 ) 2 = ( n l 1 m 2 − l 2 m 1 ) 2 l 2 ( m 1 n 2 − n 2 n 1 ) 2 = m 2 ( n 1 l 2 − n 2 l 1 ) 2 = n 2 ( l 1 m 2 − l 2 m 1 ) 2 The equations can be combined to a single equation as, l 2 + m 2 + n 2 ( m 1 n 2 − n 2 n 1 ) 2 + ( n 1 l 2 − n 2 l 1 ) 2 + ( l 1 m 2 − l 2 m 1 ) 2 Since l, m and n are the direction cosines, so, l 2 + m 2 + n 2 =1 It is known that, ( l 1 2 + m 1 2 + n 1 2 )( l 2 2 + m 2 2 + n 2 2 )−( l 1 l 2 + m 1 m 2 + n 1 n 2 )= ( m 1 n 2 − m 2 n 1 ) 2 + ( n 1 l 2 − n 2 l 1 ) 2 + ( l 1 m 2 − l 2 m 1 ) 2 ( 1 )( 1 )−0= ( m 1 n 2 − m 2 n 1 ) 2 + ( n 1 l 2 − n 2 l 1 ) 2 + ( l 1 m 2 − l 2 m 1 ) 2 ( m 1 n 2 − m 2 n 1 ) 2 + ( n 1 l 2 − n 2 l 1 ) 2 + ( l 1 m 2 − l 2 m 1 ) 2 =1 It is observed that, l 2 ( m 1 n 2 − m 2 n 1 ) 2 = m 2 ( n 1 l 2 − n 2 l 1 ) 2 = n 2 ( l 1 m 2 − l 2 m 1 ) 2 =1 On solving the above equation, the value comes out as, l= m 1 n 2 − m 2 n 1 m= n 1 l 2 − n 2 l 1 n= l 1 m 2 − l 2 m 1 Therefore, the direction cosines of the line are m 1 n 2 − m 2 n 1 , n 1 l 2 − n 2 l 1 and l 1 m 2 − l 2 m 1 .

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