Question

Find the angle between the lines whose direction ratios are proportional to 4, -3, 5 and 3, 4, 5.

• A. ${30}^o$
• B. ${60}^o$
• C. ${90}^o$
• D. $0^o$

Answer 1

Answer: (B)

${60}^o$

Let $\theta$ be the angle between the given lines.

We have,

$a_1=4,b_1=-3,c_1=5$ and $a_2=3,b_2=4,c_2=5$

Therefore,

$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

$=\frac{12-12+25}{\sqrt{16+9+25}\sqrt{9+16+25}}$

$=\frac{25}{50}=\frac{1}{2}$

$⟹\theta =\frac{\pi }{3}$

Thus, the measure of the angle between the given lines is ${60}^o$.