Using distance formula prove that the following points are collinear: `P(0,7,-7),\ Q(1,4,-5)a n d R(-1, 10 ,-9)`

Explanation

Given purposes are,
`P(0,7,-7),\ Q(1,4,-5)a n d\ R(-1,10,-9)`
we have to show that P, Q, R are collinear.
we know that if `PQ+QR=PR or PQ+PR=QR or PR+QR=PQ` then P, Q, R 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 P and Q is
`=sqrt((0-1)^2+(7-4)^2+(-7+5)^2)`
`=sqrt((-1)^2+(3)^2+(-2)^2)`
`=sqrt(1+9+4)`
`=sqrt(14)`
The distance between Q and R is
`=sqrt((1+1)^2+(4-10)^2+(-5+9)^2)`
`=sqrt((2)^2+(-6)^2+(4)^2)`
`=sqrt(4+36+16)`
`=sqrt(56)`
`=2sqrt(14)`
distance between P and R is
`=sqrt((0+1)^2+(7-10)^2+(-7+9)^2)`
`=sqrt((1)^2+(-3)^2+(2)^2)`
`=sqrt(1+9+4)`
`=sqrt(14)`
clearly we can conclude that given purposes P, Q , R are collinear as `PQ+PR=QR`