Question

A(– 2, 3, 4), B(1, 1, 2) and C(4, –1, 0) are three points. Find the Cartesian equations of the line AB and show that points A, B, C are collinear.

Answer 1

We find the cartesian equations of the line AB.

The cartesian equations of the line passing through the points (x₁, y₁, z₁) and  (x₂, y₂, z₂) are

`(x - x_1)/(x_2 - x_1) = (y - y_1)/(y_2 - y_1) = (z - z_1)/(z_2 - z_1)`

Here, (x₁, y₁, z₁) ≡ (−2, 3, 4) and (x₂, y₂, z₂) ≡ (1, 1, 2)

∴  The required cartesian equations of the line AB are

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

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

∴ `(x + 2)/(3) = (y - 3)/(-2) = (z - 4)/(-2)`

C = (4, −1, 0)

For x = 4, `(x + 2)/(3) = (4 + 2)/(3)` = 2

For y = –1, `(y - 3)/(-2) = (-1 - 3)/(-2)` = 2

For z = 0, `(z - 4)/(-2) = (0 - 4)/(-2)` = 2

∴ The coordinates of C satisfy the equations of line AB.

∴ C lies on the line passing through A and B.

Hence, A, B, C are collinear.