# If the direction ratios of a line are 5, 4, -7 respectively, then find the direction cosines.

Category: QuestionsIf the direction ratios of a line are 5, 4, -7 respectively, then find the direction cosines.
Editor">Editor Staff asked 11 months ago

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
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Editor">Editor Staff answered 11 months ago

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=&pm;$$\frac{a}{\sqrt{a^2+b^2+c^2}}$$

m=&pm;$$\frac{b}{\sqrt{a^2+b^2+c^2}}$$

n=&pm;$$\frac{c}{\sqrt{a^2+b^2+c^2}}$$

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