Question

By computing the shortest distance, determine whether following lines intersect each other.

`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5) and (x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)`

Answer 1

The shortest distance between the lines

`(x - 5)/(4) = (y -7)/(-5) = (z + 3)/(-5)`   ...(i)

`(x - 8)/(7) = (y - 7)/(1) = (z - 5)/(3)`     ...(ii)

∴  from line (i) & (ii)

 x₁ = 5, y₁ = 7, z₁ = -3, x₂ = 8, y₂ = 7, z₂ = 5

l₁ = 4, m₁ = –5, n₁ = -5, l₂ = 7, m₂ = 1, n₂ = 3

The given two lines intersect each other if and only if 

`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)| = 0`

= `|(3, 0, 8),(4, -5, -5),(7, 1, 3)|`

= 3(– 15 + 5) – 0(12 + 35) + 8(4 + 35)

= – 30 – 0 + 312

= 282

`≠` 0

∴ The given lines not intersect each other.