1.

If the direction ratios of a line are 5, 4, -7 respectively, then find the direction cosines.(a) \(\frac{75}{\sqrt{90}},\frac{4}{\sqrt{90}},\frac{5}{\sqrt{90}}\)(b) \(\frac{5}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{7}{\sqrt{90}}\)(c) \(\frac{5}{\sqrt{70}},\frac{4}{\sqrt{70}},-\frac{7}{\sqrt{70}}\)(d) \(\frac{3}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{5}{\sqrt{90}}\)This question was addressed to me in unit test.This intriguing question comes from Direction Cosines and Direction Ratios of a Line topic in portion Three Dimensional Geometry of Mathematics – Class 12

Answer»

The CORRECT option is (b) \(\frac{5}{\sqrt{90}},\frac{4}{\sqrt{90}},-\frac{7}{\sqrt{90}}\)

For explanation: If a,b,c are the direction ratios and l,m,N are the direction cosines RESPECTIVELY for a given line, then the direction cosines in terms of the direction ratios can be expressed as

l=±\(\frac{a}{\sqrt{a^2+b^2+c^2}}\)

m=±\(\frac{b}{\sqrt{a^2+b^2+c^2}}\)

n=±\(\frac{c}{\sqrt{a^2+b^2+c^2}}\)

Given that, a=5, b=4, c=-7

l=\(\frac{5}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{5}{\sqrt{(25+16+49)}}=\frac{5}{\sqrt{90}}\)

m=\(\frac{4}{\sqrt{(5^2+4^2+(-7)^2)}}=\frac{4}{\sqrt{90}}\)

n=-\(\frac{7}{\sqrt{(5^2+4^2+(-7)^2)}}=-\frac{7}{\sqrt{90}}\)


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