Question

The acute angle between two lines whose direction ratios are $2,3,6$ and $1,2,2$ is

• A. ${\cos }^{-1}\left(\frac{20}{21}\right)$
• B. ${\cos }^{-1}\left(\frac{18}{21}\right)$
• C. ${\cos }^{-1}\left(\frac{8}{21}\right)$
• D. None of these

Answer 1

The correct option is A ${\cos }^{-1}\left(\frac{20}{21}\right)$

We have $a_1=2,b_1=3,c_1=6,a_2=1,b_2=2,c_2=2$

If $\theta$ is the angle between two lines whose direction ratios are given, then

$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{{\left(a_1\right)}^2+{\left(b_1\right)}^2+{\left(c_1\right)}^2}\sqrt{{\left(a_2\right)}^2+{\left(b_2\right)}^2+{\left(c_2\right)}^2}}$

$\Rightarrow \cos \theta =\frac{2\times 1+3\times 2+6\times 2}{\sqrt{{\left(2\right)}^2+{\left(3\right)}^2+{\left(6\right)}^2}\sqrt{{\left(1\right)}^2+{\left(2\right)}^2+{\left(2\right)}^2}}=\frac{20}{21}$

$\Rightarrow \theta ={\cos }^{-1}\frac{20}{21}$