Question

Using direction ratio show that the points $A(2,3,-4),B(1,-2,3)$ and $C(3,8,-11)$ are collinear.

Answer 1

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

We have $A(2,3,-4)$ and $B(1,-2,3)$

Direction ratios are $=1-2,-2-3,3-(-4)$

$=-1,-5,7$

So,$a_1=-1,b_1=-5,c_1=7$

We have $B(1,-2,3)$ and $C(3,8,-11)$

Direction ratios are $=3-1,8-(-2),-11-3$

$=2,10,-14$

So,$a_2=2,b_2=10,c_2=-14$

Now, $\frac{a_2}{a_1}=\frac{2}{-1}=-2$

$\frac{b_2}{b_1}=\frac{10}{-5}=-2$

$\frac{c_2}{c_1}=\frac{-14}{7}=-2$

So,$\frac{a_2}{a_1}=\frac{b_2}{b_1}=\frac{c_2}{c_1}=-2$

$\therefore A,B$ and $C$ are collinear.