If a line makes an angle of 60°, 150°, 45° with the positive x, y, z-axis respectively, find its direction cosines.

(a) –\(\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\)

(b) –\(\frac{1}{2},-\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}\)

(c) \(\frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\)

(d) \(\frac{1}{2},\frac{\sqrt{3}}{2},-\frac{1}{\sqrt{2}}\)

I have been asked this question in class test.

This is a very interesting question from Direction Cosines and Direction Ratios of a Line in portion Three Dimensional Geometry of Mathematics – Class 12

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Correct answer is (c) \frac{1}{2},-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}

Easiest explanation: Let l, m, n be the direction cosines of the line.

We know that, if α, β, γ are the angles that the line makes with the x, y, z-axis respectively, then

l=cosα=cos60°=\frac{1}{2}

m=cosβ=cos150°=-\frac{\sqrt{3}}{2}

n=cosγ=cos45°=\frac{1}{\sqrt{2}}.