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

Explanation

Let the direction cosines of the required line be l, m, n
Thus, `l (-2) + m (1) + n (-1) = 0`
or ` -2l + m - n = 0` ...(i)
and `l (-3) + m (-4) + n (1) = 0`
or ` -3 l - 4m + n = 0` ...(ii)
By cross - multiplication between (i) and (ii),
` l/(1-4) = m/(3 +2) = n/(8+3)` ltbr. ` l/(-3) = m/5 = n/11`
Let us find ` sqrt(l^(2) + m^(2) + n^(2))`:
` sqrt(l^(2) + m^(2) + n^(2)) = sqrt((-3)^(2)+(5)^(2)+(11)^(2))`
` = sqrt(9 + 25 + 121) `
` = sqrt(155) `
Thus, the direction ratios of the line are , ` -3, 5, 11`
The direction cosines are :
` l = (-3)/sqrt(155) `
` m = 5/sqrt(155)`
` n = 11/sqrt(155)`