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Question
A vector `vec"r"` has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of `vec"r"`, given that `vec"r"` makes an acute angle with x-axis.
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Solution
Let `vec"a", vec"b"` and `vec"c"` three vectors such that `vec"a" = 2"k", vec"b"` = 3k and `vec"c"` = – 6k
If l, m and n are the direction cosines of vector `vec'r"`, then
l = `vec"a"/|vec"r"| = (2"k")/14 = "k"/7`
m = `vec"b"/|vec"r"| = (3"k")/14` and n = `vec"c"/|vec"r"| = (-6"k")/14 = (-3"k")/7`
We know that l2 + m2 + n2 = 1
∴ `"k"^2/49 + (9"k"^2)/196 + (9"k"^2)/49` = 1
⇒ `(4"k"^2 + 9"k"^2 + 36"k"^2)/196` = 1
⇒ 49k2 = 196
⇒ k2 = 4
∴ k = ± 2 and l = `"k"/7 = 2/7`
m = `(3"k")/14 = (3 xx 2)/14 = 3/7`
And n = `(-3"k")/7 (-3 xx 2)/7 = (-6)/7`
∴ `hat"r" = +- (2/7hat"i" + 3/7hat"j" - 6/7hat"k")`
`hat"r" = hat"r"|vec"r"|`
⇒ `vec"r" = +-(2/7hat"i" + 3/7hat"j" - 6/7hat"k")*14`
= `+- (4hat"i" + 6hat"j" - 12hat"k")`
Hence, the required direction cosines are `2/7, 3/7, (-6)/7` and the components of `vec"r"` are `4hat"i", 6hat"j"` and `-12hat"k"`.
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