If z1 and z2 are two non-zero complex numbers such that \[\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|\], then arg z1 - arg z2 is equal to :
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Official solution
Given that, \[\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|\]
\[\,\left| {{z_1} + {z_2}} \right|\] is the vector sum of z1 and z2. So \[\,\left| {{z_1} + {z_2}} \right|\] should be < \[\left| {{z_1}} \right| + \left| {{z_2}} \right|\] but here they are equal so z1 and z2 are collinear.
S if z1 makes an angle \[\theta \] with x axis then z2 will also make \[\theta \] angle.
\[\therefore\] arg z1 - arg z2 = \[\theta \] - \[\theta \] = 0