Question

The angle between a line with direction ratio $2:2:1$ and a line joining $(3,1,4)$ to $(7,2,12)$ is

• A. $co{s}^{-1}(2/3)$
• B. $co{s}^{-1}(3/2)$
• C. $ta{n}^{-1}(-2/3)$
• D. None of the above

Answer 1

Answer: (A)

Direction ratios of the line joining the points
$(3,1,4)$ and $(7,2,12)$ are
$=<7-3,2-1,12-4>$
$=<4,1,8>$
$=<a_1,a_2,a_3>$(let)
And the direction ratio of given line is
$=<2,2,1>$
$=<b_1,b_2,b_3>$(let)
Let $Q$ be the angle between the lines,
then $\cos \theta =\frac{a_1{b}_1+a_2{b}_2+a_3{b}_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}$
$\Rightarrow \cos \theta =\frac{(4)(2)+(1)(2)+(8)(1)}{\sqrt{16+1+64}\sqrt{4+4+1}}$
$\Rightarrow \cos \theta =\frac{18}{\sqrt{81}\sqrt{9}}=\frac{18}{9\times 3}=\frac{2}{3}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{2}{3}\right)$