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`.
1) – 1 / √35, 5 / √35, 3 / √35 2) 13 / √35, – 1 / √35, 1 / √35 3) 2 / √3, 5 / √3, 7 / √3 4) 3 / √35, 5 / √35, 7 / √35 Solution: (1) – 1 / √35, 5 / √35, 3 / √35
Let direction ratios of line be a, b, c
a – b + 2c = 0 — (i)
2a + b – c = 0 — (ii)
Solving (i) and (ii), we get
a = – 1, b = 5, c = 3
The direction ratios of line = (- 1, 5, 3)
The required direction cosines = – 1 / √35, 5 / √35, 3 / √35
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