Question

The angle between the lines $\frac{x+2}{2}=\frac{y+3}{2}=\frac{z-4}{1}$ and $\frac{x-1}{0}=\frac{y}{3}=\frac{z}{0}$ is

• A. ${\cos }^{-1}\left(\frac{2}{3}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• C. ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
• D. ${\cos }^{-1}\left(1\right)$

Answer 1

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

Given lines are $\frac{x+2}{2}=\frac{y+3}{2}=\frac{z-4}{1}$ and $\frac{x-1}{0}=\frac{y}{3}=\frac{z}{0}$

The direction vector of the first line is $P=2i+2j+k$ and second line $Q=3j$.

Then the cosine of the angle between the line is given by $\frac{|P\cdot Q|}{|P||Q|}$.

$\Rightarrow \cos \theta =\frac{6}{\sqrt{4+4+1}\sqrt{3^2}}$

$\Rightarrow \cos \theta =\frac{6}{3\times 3}$

$\Rightarrow \theta ={\cos }^{-1}\left(\frac{2}{3}\right)$