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A line makes equal angle with coordinate axis. Direction cosines of this line are1. ± (1, 1, 1)2. \(\pm \left(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}\right)\)3. \(\pm \left(\dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}, \dfrac{1}{\sqrt{6}}\right)\)4. \(\pm \left(\dfrac{1}{3}, \dfrac{1}{3}, \dfrac{1}{3}\right)\) |
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Answer» Correct Answer - Option 2 : \(\pm \left(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}\right)\) Concept: If a directed line L passing through the origin makes angles α, β and γ with x, y, and z-axes, respectively, called direction angles, then the cosine of these angles, namely, cosα, cosβ and cosγ are called direction cosines of the directed line L. These unique direction cosines are denoted by l, m, and n. And l2 + m2 + n2 = 1 Given: α = β = γ ∴ l = cosα m = cosβ = cosα n = cosγ = cosα \({l^2} + {m^2} + {n^2} = 1 \) \(\Rightarrow 3{\cos ^2}\alpha = 1 \) \(\Rightarrow \cos \alpha = \pm \frac{1}{{\sqrt 3 }}\) Thus the direction cosines of the line which is equally inclined to the coordinate axes are \(\pm \frac{1}{{\sqrt 3 }}, \pm \frac{1}{{\sqrt 3 }}\;and\; \pm \frac{1}{{\sqrt 3 }}\) |
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