Question

Find the angle between the two lines:

`(x + 1)/2 = (y - 2)/5 = (z + 3)/4` and `(x - 1)/5 = (y + 2)/2 = (z - 1)/(-5)`

Answer 1

Given lines are,

`(x + 1)/2 = (y - 2)/5 = (z + 3)/4`   ...(i)

And `(x - 1)/5 = (y + 2)/2 = (z - 1)/(-5)`   ...(ii)

∴ Direction ratios of the (i), (a₁, b₁, c₁) = (2, 5, 4) and direction ratios of line (ii), (a₂, b₂, c₂) = (5, 2, –5).

The angle between two lines is given as,

`cos θ = (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))`

= `(2 xx 5 + 5 xx 2 + 4 xx (-5))/(sqrt(2^2 + 5^2 + 4^2).sqrt(5^2 + 2^2 + (-5)^2)`

= `(10 + 10 - 20)/(sqrt(4 + 25 + 16).sqrt(25 + 4 + 25))`

= `0/(sqrt(45).sqrt(54))`

⇒ cos θ = 0

∴ `θ = π/2 = 90^circ`

Hence, the angle between the two lines is `π/2`.