If a line \frac{x+1}{p}=\frac{y-1}{q}=\frac{z-2}{r} where p=2q=3r, makes an angle \theta with the positive direction of y-axis, then what is \cos 2\theta equal to?

Explanation

Let p=2q=3r=6k. Thus, p=6k, q=3k, r=2k. The direction ratios of the line are (6, 3, 2). The direction cosine m with respect to the y-axis is \cos \theta = \frac{3}{\sqrt{6^2+3^2+2^2}} = \frac{3}{7}. We need \cos 2\theta = 2\cos^2\theta - 1 = 2(9/49) - 1 = \frac{18}{49} - \frac{49}{49} = -\frac{31}{49}.

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