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 .

Answer 1

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 .