If a line has the direction ratios 4, −12, 18, then find its direction cosines - Mathematics and Statistics

Sum

If a line has the direction ratios 4, −12, 18, then find its direction cosines

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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