Let O be the origin and A be the point z_1 = 1 + 2i.

Question

Let O be the origin and A be the point ${z_1} = 1 + 2i$. If B is the point ${z_2}$

Accepted Answer

$${{{z_2} - 0} \over {(1 + 2i) - 0}} = {{|OB|} \over {|OA|}}{e^{{{i\pi } \over 4}}}$$

$$ \Rightarrow {{{z_2}} \over {1 + 2i}} = \sqrt 2 {e^{{{i\pi } \over 4}}}$$

OR ${z_2} = (1 + 2i)(1 + i)$

$=  - 1 + 3i$

$$\arg {z_2} = \pi  - {\tan ^{ - 1}}3$$

$$|{z_2}| = \sqrt {10} $$

$${z_1} - 2{z_2} = (1 + 2i) + 2 - 6i = 3 - 4i$$

$$\arg ({z_1} - 2{z_2}) =  - {\tan ^{ - 1}}{4 \over 3}$$

$$|2{z_1} - {z_2}| = |2 + 4i + 1 - 3i| = |3 + i| = \sqrt {10} $$

Solutions · 1