The angle between a line with direction ratio | vedclass

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Question

(A) The direction ratios of the first line are $a_1 = 2, b_1 = 2, c_1 = 1$.The direction ratios of the second line joining the points $(3, 1, 4)$ and

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) The direction ratios of the first line are $a_1 = 2, b_1 = 2, c_1 = 1$.The direction ratios of the second line joining the points $(3, 1, 4)$ and $(7, 2, 12)$ are calculated as $(7-3, 2-1, 12-4) = (4, 1, 8)$.Let the direction ratios of the second line be $a_2 = 4, b_2 = 1, c_2 = 8$.The angle $	heta$ between the two lines is given by the formula:$\cos 	heta = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2} \sqrt{a_2^2 + b_2^2 + c_2^2}}$Substituting the values:$\cos 	heta = \frac{|2 	imes 4 + 2 	imes 1 + 1 	imes 8|}{\sqrt{2^2 + 2^2 + 1^2} \sqrt{4^2 + 1^2 + 8^2}}$$\cos 	heta = \frac{|8 + 2 + 8|}{\sqrt{4 + 4 + 1} \sqrt{16 + 1 + 64}}$$\cos 	heta = \frac{18}{\sqrt{9} \sqrt{81}} = \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 the line joining the points $(3, 1, 4)$ and $(7, 2, 12)$ is

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