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

Solution:

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

Firstly, let us calculate distance between the 3 points i.e. PQ, QR and PR.

Calculating PQ:

P = (-2, 3, 5) and Q = (1, 2, 3)

Distance = √(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

So here,

x₁ = - 2, y₁ = 3, z₁ = 5

x₂ = 1, y₂ = 2, z₂ = 3

PQ = √(1 - (- 2))² + (2 - 3)² + (3 - 5)²

= √3² + (- 1)² +(- 2)²

= √9 + 1 + 4

= √14

Calculating QR:

Q = (1, 2, 3) and R = (7, 0, - 1)

Distance = √(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

So here,

x₁ = 1, y₁ = 2, z₁ = 3

x₂ = 7, y₂ = 0, z₂ = - 1

PQ = √(7 - 1)² + (0 - 2)² + (- 1 - 3)²

= √6² + (- 2)² +(- 4)²

= √36 + 4 + 16

= √56

= 2√14

Calculating PR:

P = (-2, 3, 5) and R = (7, 0, - 1)

Distance = √(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²

So here,

x₁ = - 2, y₁ = 3, z₁ = 5

x₂= 7, y₂ = 0, z₂ = - 1

PQ = √(7 - (- 2))² + (0 - 3)² + (- 1 - 5)²

= √9² + (- 3)² +(- 6)²

= √81 + 9 + 36

= √126

= 3√14

Thus,

PQ = √14, QR = 2√14 and PR = 3√14

PQ + QR = √14 + 2√14

= 3√14

= PR

Therefore, the points P, Q and R are collinear

NCERT Solutions Class 11 Maths Chapter 12 Exercise 12.2 Question 2

Summary:

We have proved that the points (- 2, 3, 5), (1, 2, 3) and (7, 0, - 1) are collinear