Ok guys, let's say that I have this vector:
Why, when I calculate the modulus of this vector, I get $\sqrt{a^2+b^2}$, and not $\sqrt{a^2+(b\operatorname{i})^2}$. When I do the calculation in the first way I neglet $\operatorname{i}$, that is a part of a cathetus. If i is a part of a cathetus why neglet it?
If that is confusing you , you can also define the modulus $|z| $ as ;
$|z| = \sqrt{z\cdot \bar z}\qquad$$\qquad$ where $\bar z $ is the complex conjugate of $z$
let $z =a + ib$
$\bar z = a-ib$
$|z| = \sqrt{(a+ib)(a-ib)}$
$|z| = \sqrt{a^2-iab+iab-(ib)^2}$
$|z| = \sqrt{a^2-i^2\cdot b^2}$
$|z| = \sqrt{a^2+b^2}$