Question

The angle between a line whose direction ratios are in the ratio 2:3:1 and a line joining (3,1,4), to (7,2,12) is

• A. $co{s}^{-1}\left(\frac{2}{3}\right)$
• B. $co{s}^{-1}\left(\frac{-2}{3}\right)$
• C. $ta{n}^{-1}\left(\frac{2}{3}\right)$
• D. ${\cos }^{-1}\left(\frac{19}{\sqrt{1134}}\right)$

Answer 1

Answer: (D)

${\cos }^{-1}\left(\frac{19}{\sqrt{1134}}\right)$

Directions are in ratio $2:3:1$ $\Rightarrow (2k,3k,k)$ be directions.

$\therefore$ directions of line joining $(3,1,4)$ to $(7,2,12)$ is $(7-3,2-1,12-4)\equiv (4,1,8)$

$\therefore \cos \theta =\frac{2k\times 4+3k\times 1+k\times 8}{\sqrt{{\left(2k\right)}^2+{\left(3k\right)}^2+k^2}\sqrt{4^2+1^2+8^2}}=\frac{19k}{k\sqrt{14}\sqrt{81}}\Rightarrow \cos \theta =\frac{19}{\sqrt{1134}}\Rightarrow \theta ={\cos }^{-1}\left(\frac{19}{\sqrt{1134}}\right)$