Maharashtra State BoardHSC Arts 12th Board Exam
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If a line has the direction ratios 4, −12, 18, then find its direction cosines - Mathematics and Statistics

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Sum

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`.

Concept: Scalar Triple Product of Vectors
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