Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Direction ratios of a line joining $(3,1,4)$ and $(7,2,12)$ are $4,1,8$ Let $\left(a_1,b_1,c_1\right)=(2,2,1)$ and $\left(a_2,c_2\right)=(4,1,8)$. Hence angle $\theta$ between the lines is given by $$\begin{array}{rl}& \cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1\sqrt{a_2^2+b_2^2+c_2^2}}}\\ & =\frac{8+2+8}{\sqrt{4+4+1}\cdot \sqrt{16+1+64}}=\frac{18}{\sqrt{9}\cdot \sqrt{81}}=\frac{18}{(3)(9)}=\frac{2}{3}\\ & \therefore \theta ={\cos }^{-1}\left(\frac{2}{3}\right)\end{array}$$