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# Find the Direction Cosines of the Line Perpendicular to the Lines Whose Direction Ratios Are -2, 1,-1 and -3, - 4, 1 - Mathematics and Statistics

#### 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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