Question

Find the shortest distance between the lines, `(x - 1)/2 = (y - 2)/3 = (z - 3)/4` and `(x - 2)/3 = (y - 4)/4 = (z - 5)/5`.

Answer 1

The lines are

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

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

Here

x₁ = 1, y₁ = 2, z₁ = 3 and a₁ = 2, b₁ = 3, c₁ = 4

x₂ = 2, y₂ = 4, z₂ = 5 and a₂ = 3, b₂ = 4, c₂ = 5

Shortest distance between the lines is

`d = |[x_2 - x_1, y_2 - y_1, z_2 - z_1],[a_1, b_1, c_1],[a_2, b_2, c_2]|/sqrt((b_1c_2 - b_2c_1)^2 + (c_1a_2 - c_2a_1)^2 + (a_1b_2 - a_2b_1)^2)`

Now `|[x_2 - x_1, y_2 - y_1, z_2 - z_1],[a_1, b_1, c_1],[a_2, b_2, c_2]| = |[1, 2, 2],[2, 3, 4],[3, 4, 5]|`

= 1(15 – 16) – 2(10 – 12) + 2(8 – 9)

= –1 + 4 – 2

= 1

And (b₁c₂ – b₂c₁)² + (c₁a₂ – c₂a₁)² + (a₁b₂ – a₂b₁)² 

= (15 – 16)² + (12 – 10)² + (8 – 9)²

= 1 + 4 + 1

= 6

Hence, the shortest distance between lines (1) and (2) = `|1/sqrt(6)| = 1/sqrt(6)` units.

Answer 2

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 given 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 - l_1n_2)^2 + (l_1m_2 - l_2m_1)^2))|`

The equations of the given lines are

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

∴ x₁ = 1, y₁ = 2, z₁ = 3, x₂ = 2, y₂ = 4, z₂ = 5, l₁ = 2, m₁ = 3, n₁ = 4, l₂ = 3, m₂ = 4, n₂ = 5

∴ `|(x_2 - x_1, y_2 - y_1, z_2 - z_1),(l_1, m_1, n_1),(l_2, m_2, n_2)| = |(1, 2, 2),(2, 3, 4),(3, 4, 5)|`

= 1(15 – 16) – 2(10 – 12) + 2(8 – 9)

= –1 + 4 – 2

= 1

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

= (15 – 16)² + (12 – 10)² + (8 – 9)²

= 1 + 4 + 1

= 6

Hence, the required shortest distance between the given lines = `|1/sqrt(6)|` = `1/sqrt(6)` units.