Using distance formula prove that the following points are collinear: `A(4,-3,-1),\ B(5,-7,6)a n d\ C(3,1,-8)`

Explanation

Given purposes are,
`A(4,-3,-1),\ B(5,-7,6)a n d\ C(3,1,-8)`
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((4-5)^2+(-3+7)^2+(-1-6)^2)`
`=sqrt((-1)^2+(4)^2+(-7)^2)`
`=sqrt(1+16+49)`
`=sqrt(66)`
The distance between B and C is
`=sqrt((5-3)^2+(-7-1)^2+(6+8)^2)`
`=sqrt((2)^2+(-8)^2+(14)^2)`
`=sqrt(4+64+196)`
`=sqrt(264)`
`=2sqrt(66)`
distance between A and C is
`=sqrt((4-3)^2+(-3-1)^2+(-1+8)^2)`
`=sqrt((1)^2+(-4)^2+(7)^2)`
`=sqrt(1+16+49)`
`=sqrt(66)`
clearly we can conclude that given purposes A, B , C are collinear as AB+AC=BC