Question

Write the angle between the lines whose direction ratios are proportional to 1, −2, 1 and 4, 3, 2.

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{the}\mathrm{first}\mathrm{line}\mathrm{are}1,-2,1\mathrm{and}\mathrm{the}\mathrm{direction}\mathrm{ratios}\mathrm{of}\mathrm{the}\mathrm{second}\mathrm{line}\mathrm{are}4,3,2. \\ \mathrm{Let}\theta \mathrm{be}\mathrm{the}\mathrm{angle}\mathrm{between}\mathrm{these}\mathrm{two}\mathrm{lines}. \\ \mathrm{Now}, \\ \cos \theta =\left|\frac{1\left(4\right)+\left(-2\right)\left(3\right)+1\left(2\right)}{\sqrt{{\left(1\right)}^2+{\left(-2\right)}^2+{\left(1\right)}^2}\sqrt{{\left(4\right)}^2+{\left(3\right)}^2+{\left(2\right)}^2}}\right| \\ =\left|\frac{4-6+2}{\sqrt{1+4+1}\sqrt{16+9+4}}\right| \\ =\frac{0}{\sqrt{6}\sqrt{29}} \\ =0 \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{2} \\ \mathrm{Hence},\mathrm{the}\mathrm{required}\mathrm{angle}\mathrm{is}\frac{\mathrm{\pi }}{2}. \end{gathered}$$