If a line makes an angle of 120°, 45°, 30° with the positive x, y, z-axis respectively then find the direction cosines.

(a) l=\(\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)

(b) l=-\(\frac{1}{2}, \,m=-\frac{1}{\sqrt{2}}, \,n=-\frac{\sqrt{3}}{2}\)

(c) l=-\(\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)

(d) l=\(0, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)

This question was addressed to me in homework.

Question is taken from Direction Cosines and Direction Ratios of a Line topic in chapter Three Dimensional Geometry of Mathematics – Class 12

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Right answer is (c) l=-\(\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)

Easy 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

m=cosβ

n=cosγ

∴l=cos120°, m=cos45°, n=cos30°

Hence, \(l=-\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)