Question

Find the angle between the following pair of lines
$\frac{x}{2}=\frac{y}{2}=\frac{z}{1}and\frac{z-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$

Answer 1

Here, the equation of given lines are
$\frac{x}{2}=\frac{y}{2}=\frac{z}{1}and\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$
$\therefore$ Direction ratios of two lines are (2, 2, 1) and (4, 1, 8).
Let $\theta$ be the acute angle between the given lines, then
$cos\theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}\Rightarrow cos\theta =\frac{2\times 4+2\times 1+1\times 8}{\sqrt{2^2+2^2+1^2}\sqrt{4^2+1^2+8^2}}=\frac{8+2+8}{\sqrt{4+4+1}\sqrt{16+1+64}}=\frac{18}{\sqrt{9}\sqrt{81}}=\frac{18}{3\times 9}=\frac{2}{3}\Rightarrow \theta =co{s}^{-1}\left(\frac{2}{3}\right)$