Question

Show that the three lines with direction cosines are mutually perpendicular.

Answer 1

The given direction cosine of three lines are,

( l 1 , m 1 , n 1 )= 12 13 , −3 13 , −4 13

( l 2 , m 2 , n 2 )= 4 13 , 12 13 , 3 13

And,

( l 3 , m 3 , n 3 )= 3 13 , −4 13 , 12 13 .

The lines with direction cosines as l 1 , m 1 , n 1 , l 2 , m 2 , n 2 and  &  l 3 , m 3 , n 3 are perpendicular to each other when,

l 1 l 2 + m 1 m 2 + n 2 n 1 =0

Or,

l 1 l 3 + m 1 m 3 + n 3 n 1 =0

Or

l 2 l 3 + m 2 m 3 + n 3 n 2 =0

Substitute the values of ( l 1 , m 1 , n 1 ), ( l 2 , m 2 , n 2 ) and ( l 3 , m 3 , n 3 ) in above equations,

l 1 l 2 + m 1 m 2 + n 1 n 2 =( 12 13 × 4 13 )+( −3 13 × 12 13 )+( −4 13 × 3 13 ) = 48 169 + ( −36 ) 169 + ( −12 ) 169 = 48−36−12 169 =0

Thus, the two lines are perpendicular to each other.

Now,

l 2 l 3 + m 2 m 3 + n 2 n 3 =( 3 13 × 4 13 )+( −4 13 × 12 13 )+( 12 13 × 3 13 ) = 12 169 + ( −48 ) 169 + ( 36 ) 169 = 12−48+36 169 =0

Thus, the two lines are perpendicular to each other.

Also,

l 1 l 3 + m 1 m 3 + n 1 n 3 =( 12 13 × 3 13 )+( −3 13 × −4 13 )+( −4 13 × 12 13 ) = 36 169 + 12 169 + −48 169 = 12−48+36 169 =0

Thus, the two lines are perpendicular to each other.

Hence, the given three lines are mutually perpendicular.