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