Question

Using section formula, show that the points $A(2,-3,4)B(-1,2,1)$ and $C(0,\frac{1}{3},2)$ are collinear.

Answer 1

Let the points $B(-1,2,1)$ divides the join of $A(2,-3,4)$ and $C(0,\frac{1}{3},2)$ in the ratio k : 1 internally.

Then coordinates of B are

$(\frac{2}{k+1},\frac{\frac{1}{3}k-3}{k+1},\frac{2k+4}{k+1})$

Now $\frac{2}{k+1}=-1\Rightarrow 2=-k-1\Rightarrow =-3$

Thus the point B divides the join of A and C in the ratio -3 : 1 So points A, B, C are Collinear.