Question

The angle between the lines with direction ratios $(4,-3,5)$ and $(3,4,5)$ is

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

Answer: (C)

$\frac{\pi }{3}$

Answer 2

Let $\theta$ be the angle between the lines.
$\therefore$ $\cos \theta =\frac{{{a}_{1}}{{a}_{2}}+{{b}_{1}}{{b}_{2}}+{{c}_{1}}{{c}_{2}}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}$
$=\frac{4\times 3-3\times 4+5\times 5}{\sqrt{16+9+25}\sqrt{9+16+25}}$
$=\frac{1}{2}=\cos \frac{\pi }{3}$
$\Rightarrow$ $\theta =\frac{\pi }{3}$
Let the length of the side be x.