Q.Find the angle between the lines $\dfrac{x-1}{4} = \dfrac{y-3}{1} = \dfrac{z}{8}$ and $\dfrac{x-2}{2} = \dfrac{y+1}{2} = \dfrac{z-4}{1}$.

Use $\cos\theta = \dfrac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$.

Direction ratios: $(a_1,b_1,c_1) = (3,2,6)$ and $(a_2,b_2,c_2) = (-2,1,2)$.

$\cos\theta = \frac{3(-2)+2(1)+6(2)}{\sqrt{3^2+2^2+6^2}\,\sqrt{(-2)^2+1^2+2^2}} = \frac{-6+2+12}{\sqrt{9+4+36}\,\sqrt{4+1+4}} = \frac{8}{\sqrt{49}\sqrt{9}} = \frac{8}{7\times3} = \frac{8}{21}$

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Two lines are at right angles when the dot product of their direction ratios is zero.

Direction ratios of the lines:

Line 1: $(-3,\,2k,\,2)$

Line 2: $(3k,\,1,\,-5)$

For lines at right angles, the dot product of direction ratios must be zero:

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