Collinear points in Space

Question

I need help understanding how to do this question. Are (1,4,2) (4,-3,-5) (-5,-10,-8) points on the same line? Show why and how you know.

Answer 1

Let $A=(1,4,2), B=(4,-3,-5), C= (-5,-10,-8)$.

And consider the two vector: $AB=(3,-7,-7)$ $AC=(-6,-14,-10)$

$\frac{3}{-6}\not\frac{-7}{-14}\not=\frac{-7}{-10}$

So it is not at a line.

Answer 2

Hint for one possible way:

Form a triangle with three vectors from these points. Can you use the area of the triangle to determine if they are linear points?