Question

The measure of acute angle between the lines whose direction ratios are $3,2,6$ and $-2,1,2$ is __________.

• A. ${\cos }^{-1}\left(\frac{1}{7}\right)$
• B. ${\cos }^{-1}\left(\frac{8}{15}\right)$
• C. ${\cos }^{-1}\left(\frac{1}{3}\right)$
• D. ${\cos }^{-1}\left(\frac{8}{21}\right)$

Answer 1

The correct option is D ${\cos }^{-1}\left(\frac{8}{21}\right)$

Angle between the lines with direction ratios $a_1,b_1,c_1$ and $a_2,b_2,c_2$ is given by $cos\theta =\left|\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot \sqrt{a_2^2+b_2^2+c_2^2}}\right|$

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

$\cos \theta =\left|\frac{3(-2)+2\left(1\right)+6\left(2\right)}{\sqrt{3^2+2^2+6^2}\cdot \sqrt{-2^2+1^2+2^2}}\right|$

$=\left|\frac{-6+2+12}{\sqrt{9+4+36}\cdot \sqrt{4+1+4}}\right|$

$=\left|\frac{8}{\sqrt{49}\cdot \sqrt{9}}\right|$

$=\frac{8}{7\cdot 3}$

$\cos \theta =\left(\frac{8}{21}\right)$

$\theta ={\cos }^{-1}\left(\frac{8}{21}\right)\approx 67.60$