Question

Prove that the points A(3, -2, 4), B(1, 1, 1) and C(-1, 4, -2) are collinear.

Answer 1

We have

$AB=\sqrt{(1-3{)}^2+(1+2{)}^2+(1-4{)}^2}=\sqrt{4+9+9}=\sqrt{22}$

$BC=\sqrt{(-1-1{)}^2+(4-1{)}^2+(-2-1{)}^2}=\sqrt{4+9+9}=\sqrt{22}$

and $AC=\sqrt{(-1-3{)}^2+(4+2{)}^2+(-2-4{)}^2}=\sqrt{16+36+36}$

$=\sqrt{88}=2\sqrt{22}$

$\therefore AB+BC=AC$.

This shows that the given points are collinear.