Question

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

Answer 1

Given three points

$A(x_1,y_1,z_1)=(-2,3,5)$

$B(x_2,y_2,z_2)=(1,2,3)$

$C(x_3,y_3,z_3)=(7,0,-1)$

Now,

$AB=\sqrt{(x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_1-z_2{)}^2}$

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

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

$AB=\sqrt{14}$

$BC=\sqrt{(x_2-x_3{)}^2+(y_2-y_3{)}^2+(z_2-z_3{)}^2}$

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

$=\sqrt{36+4+6}$

$BC=\sqrt{56},BC=2\sqrt{14}$

$CA=\sqrt{(x_3-x_1{)}^2+(y_3-y_1{)}^2+(z_3-z_1{)}^2}$

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

$=\sqrt{81+9+36}$

$=\sqrt{126}$

$=\sqrt{2\times 3\times 3\times 7}$

$=3\sqrt{14}$

$CA=3\sqrt{14}$

now, $AB+BC=CA$

$\sqrt{14}+2\sqrt{14}=3\sqrt{14}$

Hence, the three points $A,B,C$ are collinear.