The angle between the lines frac{x}{2}=frac{y | vedclass

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(D) The direction ratios of the first line are $a_1 = 2, b_1 = 2, c_1 = 1$.The direction ratios of the second line are $a_2 = 4, b_2 = 1, c_2 = 8$.The

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$\cos ^{-1}\left(\frac{2}{3}\right)$

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(D) The direction ratios of the first line are $a_1 = 2, b_1 = 2, c_1 = 1$.The direction ratios of the second line are $a_2 = 4, b_2 = 1, c_2 = 8$.The angle $	heta$ between two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ is given by $\cos 	heta = \left| \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}} ight|$.Substituting the values: $\cos 	heta = \left| \frac{(2)(4) + (2)(1) + (1)(8)}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{4^2 + 1^2 + 8^2}} ight|$.$\cos 	heta = \left| \frac{8 + 2 + 8}{\sqrt{4 + 4 + 1} \sqrt{16 + 1 + 64}} ight| = \left| \frac{18}{\sqrt{9} \sqrt{81}} ight|$.$\cos 	heta = \frac{18}{3 	imes 9} = \frac{18}{27} = \frac{2}{3}$.Therefore,$	heta = \cos ^{-1}\left(\frac{2}{3}ight)$.

The angle between the lines $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$ is . . . . . . .

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