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Question 121 of 162

Q.If a line makes 45∘45^\circ, 60∘60^\circ with positive direction of axes xx and yy then the angle it makes with the zz-axis is :

(a) 45∘45^\circ
(b) 30∘30^\circ
(c) 60∘60^\circ
(d) 90∘90^\circ
Puducherry TnboardTamil Nadu HSC (DGE) Board 2018MCQ· 1mImportance★★★★★
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Using the direction-cosine identity cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 with the given α=45∘,β=60∘\alpha=45^\circ,\beta=60^\circ yields γ=60∘\gamma=60^\circ.

  1. If a line makes angles α,β,γ\alpha,\beta,\gamma with the positive x,y,zx,y,z axes respectively, its direction cosines cos⁡α,cos⁡β,cos⁡γ\cos\alpha,\cos\beta,\cos\gamma satisfy cos⁡2α+cos⁡2β+cos⁡2γ=1\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1.
  2. Here α=45∘\alpha=45^\circ, so cos⁡2α=(12)2=12\cos^2\alpha = \left(\dfrac{1}{\sqrt2}\right)^2=\dfrac12.
  3. And β=60∘\beta=60^\circ, so cos⁡2β=(12)2=14\cos^2\beta = \left(\dfrac12\right)^2=\dfrac14. …
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