Question

Find the Cartesian equations of the line passing through A(3, 2, 1) and B(1, 3, 1).

Answer 1

The direction ratios of the line AB are 3 – 1, 2 – 3, 1 – 1, i.e., 2, – 1, 0.

The parametric equations of the line passing through (x₁, y₁, z₁) and having direction ratios a, b, c are

x = x₁ + aλ, y = y₁ + bλ, z = z₁ + cλ

∴ the parametric equations of the line passing through (3, 2, 1) and having direction ratios 2, –1, 0 are

x = 3 + 2λ, y = 2 − λ, z = 1 + 0(λ)

∴ x – 3 = 2λ, y − 2 = −λ, z = 1

∴ `(x - 3)/(2) = (y - 2)/(-1)` = λ, z = 1

∴ The Cartesian equations of the required line are:

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