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

Category: QuestionsIf a line makes an angle of 120°, 45°, 30° with the positive x, y, z-axis respectively then find the direction cosines.
Editor">Editor Staff asked 11 months ago

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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1 Answers
Editor">Editor Staff answered 11 months ago

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=cos⁡α
 
m=cos⁡β
 
n=cos⁡γ
 
∴l=cos⁡120°, m=cos⁡45°, n=cos⁡30°
 
Hence, \(l=-\frac{1}{2}, \,m=\frac{1}{\sqrt{2}}, \,n=\frac{\sqrt{3}}{2}\)