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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)

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.

To read more, Buy study materials of Complex Numbers comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
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