The angle between a line with direction ratio | vedclass

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Question

(C) The direction ratios of the first line are $(a_1, b_1, c_1) = (2, 2, 1)$.The direction ratios of the line joining $(3, 1, 4)$ and $(7, 2, 12)$ are

Vedclass · Accepted Answer

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

Vedclass · Answer

(C) The direction ratios of the first line are $(a_1, b_1, c_1) = (2, 2, 1)$.The direction ratios of the line joining $(3, 1, 4)$ and $(7, 2, 12)$ are $(a_2, b_2, c_2) = (7-3, 2-1, 12-4) = (4, 1, 8)$.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 $\cos 	heta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$.Substituting the values,we get $\cos 	heta = \frac{|(2)(4) + (2)(1) + (1)(8)|}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{4^2 + 1^2 + 8^2}}$.$\cos 	heta = \frac{|8 + 2 + 8|}{\sqrt{4 + 4 + 1} \sqrt{16 + 1 + 64}} = \frac{18}{\sqrt{9} \sqrt{81}}$.$\cos 	heta = \frac{18}{3 	imes 9} = \frac{18}{27} = \frac{2}{3}$.Therefore,$	heta = \cos ^{-1}\left(\frac{2}{3}ight)$.

The angle between a line with direction ratios $2, 2, 1$ and a line joining $(3, 1, 4)$ and $(7, 2, 12)$ is

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