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Question
If a line has the direction ratios 4, −12, 18, then find its direction cosines
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Solution
Direction ratios of the line are a = 4, b = −12, c = 18.
Let l, m, n be the direction cosines of the line.
Then l = `"a"/sqrt("a"^2 + "b"^2 + "c"^2)`
= `4/(sqrt(4^2 + (-12)^2 + (18)^2))`
= `4/(sqrt(16 + 144 + 324))`
= `4/22`
= `2/11`
m = `"b"/(sqrt("a"^2 + "b"^2 + "c"^2))`
= `(-12)/sqrt(4^2 + (-12)^2 + (18)^2)`
= `(-12)/sqrt(16 + 144 + 324)`
= `(-12)/22`
= `(-6)/11`
and
n = `"c"/sqrt("a"^2 + "b"^2 + "c"^2)`
= `18/sqrt(4^2 + (-12)^2 + (18)^2)`
= `18/(sqrt(16 + 144 + 324))`
= `18/22`
= `9/11`
Hence, the direction cosines of the line are `2/11, (-6)/11, 9/11`.
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