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Question
The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are ______.
Options
4, 5, 7
4, –5, 7
4, –5, –7
–4, 5, 8
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Solution
The direction ratios of the line which is perpendicular to the two lines `(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1) and (x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)` are 4, 5, 7.
Explanation:
`(x - 7)/(2) = (y + 17)/(-3) = (z - 6)/(1)`
`(x + 5)/(1) = (y + 3)/(2) = (z - 4)/(-2)`
The direction ratios of the given lines are proportional to 2, -3, 1 and 1, 2, -2.
The given lines are parallel to the vectors →
`vecb_1 = 2hati - 3hatj + hatk and vecb_2 = hati + 2hatj - 2 hatk`
The vector perpendicular to the given two lines is →
`vecb = vecb_1 xx vecb_2`
= `|(hati hatj hatk), (2 -3 1), (1 2 -2)|`
= `4hati + 5hatj + 7hatk`
Hence, the direction ratios of the line perpendicular to the given two lines are proportional to 4, 5, 7.
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