Unit Vectors

Unit vectors are vectors of norm.

$$\hat{v}=\frac{1}{\Vert v\Vert }\cdot v\text{in }{\mathbb{R}}^n$$

complex vectors

$$\Vert |v_1⟩\Vert =\left[\begin{array}{c}\frac{1}{\sqrt{2}}\\ \frac{-i}{\sqrt{2}}\end{array}\right]={\left|\frac{1}{\sqrt{2}}\right|}^2+{\left|\frac{-i}{\sqrt{2}}\right|}^2=\frac{1}{2}+\frac{1}{2}=1$$ $$|v_2⟩=\left[\begin{array}{c}\frac{1+i}{2}\\ \frac{1-i}{2}\end{array}\right]={\left(\frac{1+i}{2}\right)}^2+{\left|\frac{1-i}{2}\right|}^2$$ $$=\left[{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2\right]+\left[{\left(\frac{1}{2}\right)}^2+{\left(\frac{1}{2}\right)}^2\right]=1$$ $$|v⟩=\left[\begin{array}{c}\alpha \\ \beta \end{array}\right],\alpha ,\beta \in \mathbb{C}\Rightarrow |\alpha {|}^2+|\beta {|}^2=1\text{ for }|v⟩\text{ to be a unit vector}$$

Distance

Given $P(u_1,u_2)$ and $Q(v_1,v_2)$:

$$|PQ|=\sqrt{(v_1-u_1{)}^2+(v_2-u_2{)}^2}$$

Given two vectors $u$ and $v$ in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$, the distance is defined as $d(u,v)=\Vert u-v\Vert$