Question

Direction ratios of two lines are $5,-12,13$ and $-3,4,5,$ then angle between the lines is

• A. ${\cos }^{-1}\left(\frac{2}{65}\right)$
• B. ${\cos }^{-1}\left(\frac{1}{65}\right)$
• C. ${\cos }^{-1}\left(\frac{3}{65}\right)$
• D. $\frac{\pi }{3}$

Answer 1

Answer: (B)

Direction ratios of both lines are
$a_1=5,b_1=-12,c_1=13$ and $a_2=-3,b_2=4,c_2=5,$
then angle between them is given by
$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
$=\frac{5\times -3+4\times -12+13\times 5}{\sqrt{{(5)}^2+{(-12)}^2+{(13)}^2}\sqrt{{(-3)}^2+{(4)}^2+{(5)}^2}}$
$=\frac{-15-48+65}{\sqrt{25+144+169}\sqrt{9+16+25}}$
$=\frac{2}{13\sqrt{2}.5\sqrt{2}}$
$\Rightarrow$ $\cos \theta =\frac{1}{65}$
$\Rightarrow$ $\theta ={\cos }^{-1}\left(\frac{1}{65}\right)$