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# If a line makes angles $90^\circ$,$135^\circ$, $45^\circ$ with the x, y and z-axes respectively, find its direction cosines.

$\begin{array}{1 1} (0,\large\frac{1}{\sqrt 2},1) \\ (0,\large\frac{-1}{\sqrt 2},-1) \\ (0,\large\frac{-1}{\sqrt 2},1) \\ (0,\large\frac{-1}{ 2},1)\end{array}$

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• The cosine of the angles made by the directed line, passing through the orgin with the x,y and z axes are called direct cosines.
Given that $\alpha =90^{\circ},\:\:\beta=135^{\circ}\:\:and\:\:\gamma=45^{\circ}$
The direction cosines are
$\cos \alpha =\cos 90^{\circ}=0$
$\cos \beta =\cos 135^{\circ}=\cos(180^{\circ}-45^{\circ})=\large\frac{-1}{\sqrt 2}$
$\cos \gamma =\cos 45^{\circ}=1$
Hence the direction cosines are $(0,\large\frac{-1}{\sqrt 2}$$,1)$
answered Jun 3, 2013 by