Question

The measure if the angle between the lines x=k+1, y=2k-1, z=2k+3, K ∈ R and x-1/2 = y+1/1 = z-1/-2 is.........,Select the correct option from the given options.
(a) sin⁻¹ 4/3
(b) cos⁻¹ 4/9
(c) sin⁻¹ √5/3
(d) π/2

Answer 1

for solution refer attachment and option d is correct

Answer 2

given, equation of lines are :
x = k + 1 , y = 2k - 1 , z = 2k + 3
it can be written as (x - 1)/1 = (y + 1)/2 = (z - 3)/2
and (x - 1)/2 , (y + 1)/1 = (z - 1)/-2

here, $a_1=1,b_1=2,c=2$
and $a_2=2,b_2=1,c_2=-2$

use formula of angle between two lines ,
e.g., $\theta=cos^{-1}\frac{a_1a_2+b_1b_2+c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

$\theta=cos^{-1}\frac{1\times2+2\times1+2\times-2}{\sqrt{1^2+2^2+2^2}\sqrt{2^2+1^2+(-2)^2}}$

$\theta=cos^{-1}\frac{2+2-4}{\sqrt{9}\sqrt{9}}$

$\theta=cos^{-1}0$

$\theta=\frac{\pi}{2}$

hence, option (d) is correct.