Question

Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and (0,13,2)are collinear.

Answer 1

The given points are A (2, –3, 4), B (–1, 2, 1), and (0,1/3,2).
Let P be a point that divides AB in the ratio k:1.
Hence, by section formula, the coordinates of P are given by

Now, we find the value of k at which point P coincides with point C.
By taking -k+2 / k+1=0 we obtain k = 2.
For k = 2, the coordinates of point P are (0,1/3,2).
i.e., (0,1/3,2) is a point that divides AB externally in the ratio 2:1 and is the same as point P.
Hence, points A, B, and C are collinear.