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

Every point of space is located by an ordered triple $(x,y,z)$, its signed perpendicular distances from the three coordinate planes. Applying the Pythagorean theorem twice -- first across a coordinate plane, then out of it -- gives the distance between two points $P_1(x_1,y_1,z_1)$ and $P_2(x_2,y_2,z_2)$ as $P_1P_2=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$, the natural 3D extension of the 2D distance formula. This single f …