Question

If $A(-2,3,4)$, $B\equiv (1,1,2)$ and $C\equiv (4,-1,0)$ are three points, then prove that the points$A,B,C$ are collinear.

Answer 1

Given ponts are $A(-2,3,4)B(1,1,2)andC(4,-1,0)$

Three point are collinear.If direction ratios of

$AB$ and $BC$ are proportional

$AB$ $BC$

$A(-2,3,4)$ $B(1,1,2)C(4,-1,0)$

$B(1,1,2)$ Directin ratios

Direction ratios $=4-1,-1-1,0-2$

$=1-(-2),+1-3,2-4$ $=-3,-2,-2$

$=3,-2,-2$

so, $a_1=1,b_1=-2,c_1=-2$ $a_2=3,b_2=-2,c_2=-2$

Now, $\frac{a_1}{a_2}=\frac{b3}{3}=1$

$\frac{b_1}{b_2}=\frac{-2}{2}=1$

$\frac{c_1}{c_2}=\frac{-2}{{}_2}=1$

$\therefore A,B,C$ are collinear