Question

The angle between the lines, Whose direction ratios are $\left(1,1,2\right)$, $\left(\sqrt{3}-1,-\sqrt{3}-1,4\right)$ is

• A. ${\cos }^{-1}\left(\frac{1}{65}\right)$
• B. $\frac{\mathrm{\pi }}{6}$
• C. $\frac{\mathrm{\pi }}{3}$
• D. $\frac{\mathrm{\pi }}{2}$

Answer 1

The correct option is C

$\frac{\mathrm{\pi }}{3}$

Explanation for correct answer Option(s)

Find the angle between the lines.

The given points are,

$\left(1,1,2\right)$, and $\left(\sqrt{3}-1,-\sqrt{3}-1,4\right)$

We know that the angles between the two lines is given by:

$\cos \theta =\frac{x_1{x}_2+y_1{y}_2+z_1{z}_2}{\sqrt{x_1^2+y_1^2+z_1^2}\sqrt{x_2^2+y_2^2+z_2^2}}$

Substitute, $x_1=1,y_1=1,z_1=2,x_2=\sqrt{3}-1,y_2-\sqrt{3}-1,z_2=4$ in the above Equation

$$\begin{gathered} \cos \theta =\frac{1\left(\sqrt{3}-1\right)+1\left(-\sqrt{3}-1\right)+2\left(4\right)}{\sqrt{1^2+1^2+2^2}\sqrt{{\left(\sqrt{3}-1\right)}^2+{\left(-\sqrt{3}-1\right)}^2+4^2}} \\ \cos \theta =\frac{\sqrt{3}-1-\sqrt{3}-1+8}{\sqrt{1+1+4}\sqrt{3+1-2\sqrt{3}+3+1+2\sqrt{3}+16}} \\ \cos \theta =\frac{8-2}{\sqrt{6}\sqrt{24}} \\ \cos \theta =\frac{6}{\sqrt{144}} \\ \cos \theta =\frac{6}{12} \\ \cos \theta =\frac{1}{2} \\ \therefore \theta ={\cos }^{-1}\left(\frac{1}{2}\right) \\ \theta =\frac{\mathrm{\pi }}{3} \end{gathered}$$

Hence, the angle between the lines is $\frac{\mathrm{\pi }}{3}$

Therefore, the correct answer is Option C.