Question

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 1

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