Show that the points (–2, 3, 5), (1, 2, 3) and (7, 0, –1) are collinear.

If three points are collinear, then they lie on a line.

Let first calculate distance between the 3 points

i.e. PQ, QR and PR

Calculating PQ

P ≡ (– 2, 3, 5) and Q ≡ (1, 2, 3)

Distance PQ

Here,

x₁ = – 2, y₁ = 3, z₁ = 5

x₂ = 1, y₂ = 2, z₂ = 3

Distance PQ

Calculating QR

Q ≡ (1, 2, 3) and R ≡ (7, 0, – 1)

Distance QR

Here,

x₁ = 1, y₁ = 2, z₁ = 3

x₂ = 7, y₂ = 0, z₂ = – 1

Distance QR

Calculating PR

P ≡ (– 2, 3, 5) and R ≡ (7, 0, – 1)

Distance PR

Here,

x₁ = – 2, y₁ = 3, z₁ = 5

x₂ = 7, y₂ = 0, z₂ = – 1

Distance PR

Thus, PQ = , QR = & PR =

So, PQ + QR = + = = PR

Thus, Points P, Q and R are collinear.