Question

If direction cosines of two lines are proportional to $(2,3,-6)$ and $(3,-4,5)$, then the acute angle between them is

• A. ${\cos }^{-1}\left(\frac{49}{36}\right)$
• B. ${\cos }^{-1}\left(\frac{18\sqrt{2}}{35}\right)$
• C. ${96}^o$
• D. ${\cos }^{-1}\left(\frac{18}{35}\right)$

Answer 1

The correct option is B ${\cos }^{-1}\left(\frac{18\sqrt{2}}{35}\right)$

We know that
$\cos \theta =\frac{|a_1{a}_2+b_1{b}_2+c_1{c}_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$
$\therefore \cos \theta =\frac{\left|\right(2\left)\right(3)+(3\left)\right(-4)+(-6\left)\right(-5\left)\right|}{\sqrt{2^2+3^2+{(-6)}^2}\sqrt{3^2+{(-4)}^2+5^2}}$
$=\frac{|6-12-30|}{\sqrt{4+9+36}\sqrt{9+16+25}}$
$\Rightarrow \cos \theta =\frac{36}{7.5\sqrt{2}}=\frac{18\sqrt{2}}{35}$
$\Rightarrow \theta ={\cos }^{-1}\left(\frac{18\sqrt{2}}{35}\right)$