Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Direction ratios of the line joining the points $(3,1,4)$ and $(7,2,12)$ are $$\begin{array}{l}=⟨7-3,2-1,12-4⟩\\ =<4,1,8>\\ =⟨a_1,a_2,a_3⟩\end{array}$$ And the direction ratio of given line is $$\begin{array}{l}=<2,2,1>\\ =<b_1,b_2,b_3>\end{array}$$ Let $Q$ be the angle between the lines, $$\text{ then }\cos \theta =\frac{a_1{b}_1+a_2{b}_2+a_3{b}_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}$$ $$\begin{array}{l}\Rightarrow \cos \theta =\frac{(4)(2)+(1)(2)+(8)(1)}{\sqrt{16+1+64}\sqrt{4+4+1}}\\ \Rightarrow \cos \theta =\frac{18}{\sqrt{81}\sqrt{9}}=\frac{18}{9\times 3}=\frac{2}{3}\\ \Rightarrow \theta ={\cos }^{-1}\left(\frac{2}{3}\right)\end{array}$$