Question

The angle between a line with direction ratio $2:2:1$ and a line joining $(3,1,4)$ to $(7,2,12)$ is

• A. ${\cos }^{-1}\frac{2}{3}$
• B. ${\cos }^{-1}\frac{3}{2}$
• C. ${\tan }^{-1}-\frac{2}{3}$
• D. None of these

Answer 1

The correct option is A ${\cos }^{-1}\frac{2}{3}$

Direction ratio of the line joining $(3,1,4)$ and $(7,2,12)$ be $\left(\right(7-3),(2-1),(12-4\left)\right)$ i.e. $(4,1,8).$

Let angle between the two be $\theta .$

$(2\hat{i}+2\hat{j}+\hat{k}).(4\hat{i}+\hat{j}+8\hat{k})=|2\hat{i}+2\hat{j}+\hat{k}||4\hat{i}+\hat{j}+8\hat{k}|\cos \theta$

$\Rightarrow 8+2+8=3\times 9\cos \theta$

$\Rightarrow \cos \theta =\frac{18}{27}=\frac{2}{3}$

$\Rightarrow \theta ={\cos }^{-1}\frac{2}{3}$

Hence, A is the correct option.