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# Find the angle between the following lines. $\large\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-4}{6}$ and $x+1=\large\frac{y+2}{2}=\frac{z-4}{2}$

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The D.r.s of the lines $\large\frac{x-1}{2} = \large\frac{y+1}{3}= \large\frac{z-4}{2}$ and $x+1 = \large\frac{y+2}{2} = \large\frac{z-4}{2}$ are $(a_1, b_1, c_1) = (2, 3, 6), (a_2, b_2, c_2 ) = (1, 2, 2)$
The angle between them = $\cos^{-1} \bigg[ \large\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}} \bigg]$
$\cos^{-1} \bigg( \large\frac{2+6+12}{\sqrt{4+9+36}\sqrt{1+4+4}} \bigg)$
$\cos^{-1} \large\frac{20}{(7)(3)} = \cos^{-1} \large\frac{20}{21}$
answered Jun 12, 2013

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