Question

Show that the points (2,3,4),(-1,-2,1),(5,8,7)are collinear.

Answer 1

The given points are A(2,3,4),B(-1,-2,1),and C(5,8,7).
It is known that the direction ratio of line joining the points(x1,y1,z1),and(x2,y2,z2),
are given by,x2-x1,y2-y1,and z2-z1
The direction ratios of AB are(-1 -2),(-2-3),and(1-4)i.e.,-3,-5,and-3.
The direction ratios of BC are(5-(-1)),(8-(-2)),and(7,-1)i.e.,6,10,and 6.
It can be seen that the direction ratios of BC are -2 times that of AB i.e.,they are proportional.
Therefore,AB is parallel to BC.Since point B is common to both AB and BC,points A,B,and C are collinear.