Using distance formula prove that the following points are collinear: `A(3,-5,1), B(-1,0,8)\ a n d C(7,-10 ,-6)`

Explanation

Given purposes are,
`A(3,-5,1),\ B(-1,0,8)a n d\ C(7,-10,-6)`
we have to show that A, B, C are collinear.
we know that if `AB+BC=AC or AB+AC=BC or AC+BC=AB` then A, B, C are said to be collinear.
By using distance formula,
The distance between any two points (a, b, c) and (k, l, m) is given by,
`sqrt((a-k)^2+(b-l)^2+(c-m)^2)`
so ,distance between A and B is
`=sqrt((3+1)^2+(-5-0)^2+(1-8)^2)`
`=sqrt((4)^2+(-5)^2+(-7)^2)`
`=sqrt(16+25+49)`
`=sqrt(90)`
`=3sqrt(10)`
The distance between B and C is
`=sqrt((-1-7)^2+(0+10)^2+(8+6)^2)`
`=sqrt((-8)^2+(10)^2+(14)^2)`
`=sqrt(64+100+196)`
`=sqrt(360)`
`=6sqrt(10)`
distance between A and C is
`=sqrt((3-7)^2+(-5+10)^2+(1+6)^2)`
`=sqrt((-4)^2+(5)^2+(7)^2)`
`=sqrt(16+25+49)`
`=sqrt(90)`
`=3sqrt(10)`
clearly we can conclude that given purposes A, B , C are collinear as AB+AC=BC