Class 11 Mathematics Q3 (Quadratic Equation):

Q3. If $z_1$ and $z_2$ are two non-zero complex numbers such that

$$\mathrm{\mid }{z}_1+z_2\mathrm{\mid }=\mathrm{\mid }{z}_1\mathrm{\mid }+\mathrm{\mid }{z}_2\mathrm{\mid },$$

then find $\arg (z_1)-\arg (z_2)$.

Step 1:

For any complex numbers (or vectors), we know:

$$\mathrm{\mid }{z}_1+z_2\mathrm{\mid }\le \mathrm{\mid }{z}_1\mathrm{\mid }+\mathrm{\mid }{z}_2\mathrm{\mid }$$

Equality holds only when $z_1$ and $z_2$ are in the same direction (i.e., they have the same argument/angle).

Step 2:

Given,

$$\mathrm{\mid }{z}_1+z_2\mathrm{\mid }=\mathrm{\mid }{z}_1\mathrm{\mid }+\mathrm{\mid }{z}_2\mathrm{\mid }$$

This means equality holds, so $z_1$ and $z_2$ must lie on the same straight line from the origin in the complex plane, pointing in the same direction.

Step 3:

If two complex numbers point in the same direction, then,

$$\arg (z_1)=\arg (z_2)$$

So,

$$\arg (z_1)-\arg (z_2)=0$$

Answer: 0