Question

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

Answer 1

Let points $(-2,3,5),(1,2,3)$ and $(7,0,-1)$ be denoted by $P,Q$ and $R$ respectively.

Points $P,Q$ and $R$ are collinear if they lie on a line.
$PQ=\sqrt{{(1+2)}^2+{(2-3)}^2+{(3-5)}^2}$
$=\sqrt{{\left(3\right)}^2+{(-1)}^2+{(-2)}^2}$
$=\sqrt{9+1+4}$
$=\sqrt{14}$
$QR=\sqrt{{(7-1)}^2+{(0-2)}^2+{(-1-3)}^2}$
$=\sqrt{{\left(6\right)}^2+{(-2)}^2+{(-4)}^2}$
$=\sqrt{36+4+16}$
$=\sqrt{56}$
$=2\sqrt{14}$
$PR=\sqrt{{(7+2)}^2+{(0-3)}^2+{(-1-5)}^2}$
$=\sqrt{{\left(9\right)}^2+{(-3)}^2+{(-6)}^2}$
$=\sqrt{81+9+36}$
$=\sqrt{126}$
$=3\sqrt{14}$

Here $PQ+QR=\sqrt{14}+2\sqrt{14}$
$=3\sqrt{14}$
$=PR$
$\Rightarrow PQ+QR=PR$

Hence points $P(-2,3,5),Q(1,2,3)$ and $R(7,0,-1)$ are collinear