Question

Find the direction cosines of the line perpendicular to the lines whose direction ratios are -2, 1,-1 and -3, - 4, 1 

Answer 1

Let `bara   "and"   barb` be the vectors along the lines whose direction ratios are -2, 1, -1 and -3, -4, 1 respectively.

∴ `bara = -2hati + hatj - hatk and hatb = -3hati - 4hatj + hatk`

A vector perpendicular to both `bara and barb` is given by

`bara xx barb = |(hati, hatj, hatk), (-2, 1, -1), (-3, -4, 1)|`

= `( 1 - 4 )hati - ( - 2 - 3 )hatj + ( 8 + 3 )hatk`

= `-3hati + 5hatj + 11hatk`

∴ the direction ratios of the required line are -3, 5, 11
Now, `sqrt( 9 + 25 + 12)  = sqrt155`

Direction cosine of the line are `-3/sqrt155, 5/sqrt155, 11/sqrt155`.