Question

Show that the points $(2,-1,3),(3,-5,1)$ and $(-1,11,9)$ are collinear.

Answer 1

Given points, $(2,-1,3),(3,-5,1)$ and $(-1,11,9)$

Three points $A,B,C$ are collinear if direction ratios of AB and BC are proportional

$A\equiv (2,-1,3);B\equiv (3,-5,1);C\equiv (-1,11,9)$

$AB\to$ Direction ratios $\equiv (1,-4,-2)$

$BC\to$ Direction ratios $\equiv (-4,16,8)$

$\therefore a_1=1,b_1=-4,c_1=-2$

$a_2=-4,b_2=16,c_2=8$

$\therefore \frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}=\frac{-1}{4}$

Therefore $A,B,C$ are collinear