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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)
By using the 3d distance formula,
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
Show that the points (- 2, 3, 5), (1, 2, 3) and (7, 0, - 1) are collinear
Summary:
We have proved that the points (- 2, 3, 5), (1, 2, 3) and (7, 0, - 1) are collinear
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