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Distance, Triangle Inequality

 

If z1 = x1 + iy1, z2 = x2 + iy2, then distance between points z1, z2 is argard plane is |z1-z2|= √((x1-x2)2 + (y1-y2)2)

distance-triangle-inequality 

In triangle OAC,

                OC ≤ OA + AC

                OA ≤  AC + OC

                AC ≤ OA + OC

Using these inequalities we have

      ||z1| - |z2|| ≤ |z1+z2| ≤ |z1| + |z2|

Similarly from triangle OAB, we have

      ||z1| - |z2|| ≤ |z1-z2| ≤ |z1| + |z2|

Note:        

 (a)  ||z1| - |z2|| = |z1+z2| , |z1-z2| = |z1| + |z2|  iff origin, z1, and z2 are collinear and origin lies between z1 and z2.

 (b)  |z1 + z2| = |z1|+|z2| , ||z1| - |z2|| = |z1-z2| iff origin, z1 and z2 are collinear and z1 and z2 lies on the same side of origin.

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