Question

The angle between the lines, whose direction ratios are $1,1,2$ and $\sqrt{3}-1,-\sqrt{3}-1,4,$ is

• A. ${45}^{\circ }$
• B. ${30}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is C ${60}^{\circ }$

The direction ratios are $(1,1,2)$ and $(\sqrt{3}-1,-\sqrt{3}-1,4)$

Angle between them is given by

$\cos \theta =\frac{1(\sqrt{3}-1)+1(-\sqrt{3}-1)+2/4)}{\sqrt{1^2+1^2+2^2}\sqrt{(\sqrt{3}-1{)}^2+(-\sqrt{3}-1{)}^2+4^2}}$

$=\frac{\sqrt{3}-1-\sqrt{3}-1+8}{\sqrt{6}\sqrt{8+16}}$

$=\frac{6}{\sqrt{6}\sqrt{24}}=\frac{6}{\sqrt{144}}$

$=\frac{6}{12}$

$=\frac{1}{2}$

$\cos \theta =\frac{1}{2}$

$\Rightarrow \theta =\frac{\pi }{3}={60}^o$.