# What is the sum of squares of direction cosines of all the four diagonals of the cube?

Solution: If$\\(l,m,n)$ be the direction cosine of a line

which makes angles $\\\alpha ,\beta ,\gamma ,\delta$ with these four diagonals of the cube , then

$\\ cos\alpha =(l+m+n)/\sqrt{3} \\ \\cos\beta =(-l+m+n)/\sqrt{3}\\ \\cos\gamma =(l-m+n)/\sqrt{3}\\ \\cos\delta =(l+m-n)/\sqrt{3}$

Therefore ,$\\ cos^2\alpha +cos^2\beta +cos^2\gamma +cos^2\delta =4/3$

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