Question

Find the shortest distance between the lines `(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`

Answer 1

The shortest distance between the lines

`(x - x_1)/(l_1) = (y - y_1)/(m_1) = (z - z_1)/(n_1) and (x - x_2)/(l_2) = (y - y_2)/(m_2) = (z - z_2)/(n_2) ` is give n by

d = `||(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)|/sqrt((m_1n_2 - m_2n_1)^2 + (l_2n_1 - 1_1n_2)^2 + (l_1m_2 - l_2m_1)^2)|`

The equation of the given lines are

`(x + 1)/(7) = (y + 1)/(-6) = (z + 1)/(1) and (x - 3)/(1) = (y - 5)/(-2) = (z - 7)/(1)`

∴ x₁ = –1, y₁ = – 1, z₁ = – 1, x₂ = 3, y₂ = 5, z₂ = 7,

l₁ = 7, m₁ = – 6, n₁ = 1, l₂ = 1, m₂ = – 2, n₂ = 1

`|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)| = |(4, 6, 8),(7, -6, 1),(1, -2, 1)|`

= 4(– 6 + 2) – 6(7 – 1) + 8(– 14 + 6)

= – 16 – 36 – 64

= – 116

and (m₁n₂ – m₂n₁)² + (l₂n₁ – l₁n₂)² + (l₁m₂ – l₂m₁)²  

= (– 6 + 2)² + (1 – 7)² + (– 14 + 6)²

= 16 + 36 + 64

= 116

Hence, the required shortest distance between the given lines

= `|(-116)/sqrt(116)|`

= `sqrt(116)`

= `2sqrt(29)  "units"`.