Question

1. Show that the points (5,4, 2), (6, 2, -1) and (8, -2, –7) are collinear.​

Answer 1

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Answer 2

Given

we have given three points say A ( 5,4,2),B(6,2,-1) & C( 8,-2,-7)

To show

we have to show that there points are collinear

How to do

We will find distance between AB ,BC and CA and hence we will show if AB+BC =CA then the given points are collinear and vice -versa.

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

Formula used here : Distance formula(D)

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

We will find distance between AB

A(5,4,2) and B(6,2,-1)

x₁=5 ;x₂ = 6 ; y₁= 4 & y₂ = 2

z₁=2 & z₂=-1

AB= √(6-5)²+(2-4)²+(-1-2)²

AB=√1+4+9=√14

AB=√14 units

Distance between BC

B(6,2,-1) & C=(8,-2,-7)

x₁=6 ;x₂ = 8 ; y₁= 2 & y₂ = -2

z₁=-1 & z₂=-7

BC=√(8-6)²+(-2-2)²+(-7+1)²

BC=√(2)²+(-4)²+(-6)²

BC=√4+16+36

BC=√20+36=√56=2√14

BC=2√14 units

Distance between CA

A(5,4,2) & C(8,-2,-7)

x₁=5;x₂ = 8; y₁= 4 & y₂ = -2

z₁=2 & z₂=-7

CA=√(8-5)²+(-2-4)²+(-7-2)²

CA=√3²+(-6)²+(-9)²

CA=√9+36+81

CA=√126=3√14

CA=3√14

Here, AB+BC=CA

=>√14+2√14=3√14

√14+2√14=3√14 Hence,proved

So, The points A(5,4,2),B(6,2,-1) & C(8,-2,-7) are collinear.