Question

Show that the points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ are collinear.

Answer 1

Let $A(-2,3,5),B(1,2,3)$ and $C(7,0,-1)$ be three given points.

Then

$AB=\sqrt{(1+2{)}^2+(2-3{)}^2+(3-5{)}^2}$

$=\sqrt{9+1+4}=\sqrt{14}$

$BC=\sqrt{(7-1{)}^2+(0-2{)}^2+(-1-3{)}^2}$

$=\sqrt{36+4+16}=\sqrt{56}=2\sqrt{14}$

$AC=\sqrt{(7+2{)}^2+(0-3{)}^2+(-1-5{)}^2}$

$=\sqrt{81+9+36}=\sqrt{126}=3\sqrt{14}$

Now $AC=AB+BC$

Thus, A, B and C are collinear points.