# Are the points (2, 3, 4), (-1, -2, 1), (5, 8, 7) collinear?

Toolbox:
• If the direction ratios of two lines segements are proportional, then the lines ae parallel.
Given $A(2,3,4),B(-1,-2,1),C(5,8,7)$
Direction ratio of line joining $A$ and $B$ are
$(-1-2),(-2-3),(1-4)$
$=(-3,-5,-3)$
Direction ratio of line joining $B$ and $C$ are
$(5-(-1)),(8-(-2)),(7-1)$
$=(6,10,6)$
It is clear that the direction ration of $AB$ and $BC$ are proportional, hence $AB$ is parallel to $BC$ .
But $B$ is common to both $AB$ and $BC$
Therefore $A,B,C$ are proportional and collinear.