Question

Find the acute angle between the lines in $3$-dimensional space, whose direction ratios are proportional to $6,9,18$ and $1,2,2.$

• A. ${\cos }^{-1}\left(\frac{19}{21}\right)$
• B. ${\sin }^{-1}\left(\frac{19}{21}\right)$
• C. ${\cos }^{-1}\left(\frac{20}{21}\right)$
• D. ${\sin }^{-1}\left(\frac{20}{21}\right)$

Answer 1

Answer: (C)

${\cos }^{-1}\left(\frac{20}{21}\right)$

The drs of two lines are proportional to $6,9,18$ and $1,2,2$

Let $a=6i+9j+18k$ and $b=i+2j+2k$

The angle between the lines is ${\cos }^{-1}\left(\frac{a.b}{|a||b|}\right)={\cos }^{-1}\left(\frac{6+18+36}{21\times 3}\right)={\cos }^{-1}\left(\frac{20}{21}\right)$