Norm And Unit Vectors

Norm of Real vectors: a geometric vector is characterised by two properties, magnitude and direction.

Magnitude: length in Euclidean Geometry. $v=(v_1,v_2)$ in ${\mathbb{R}}^2$. The length can be calculated using Pythagorean theorem.

$$\Vert v\Vert =\sqrt{v_1^2+v_2^2}$$

The length of $v$ is a scalar denoted as $\Vert v\Vert$, known as “norm of $v$.”

Why do mathematicians need complex words for simple things? Norm $\equiv$ length $\equiv$ magnitude.

Important: norm can be extended to generalise vectors such as matrices and functions

In ${\mathbb{R}}^3$:

$$\Vert v\Vert =\sqrt{v_1^2+v_2^2+v_3^2}$$

Eg.

$$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)\Rightarrow \sqrt{{\left(\frac{\sqrt{2}}{2}\right)}^2+{\left(\frac{\sqrt{2}}{2}\right)}^2}=\sqrt{\frac{1}{2}+\frac{1}{2}}=1$$ $$\left(\frac{1}{3},\frac{1}{3},\frac{1}{3}\right)=\sqrt{{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{3}\right)}^2}=\sqrt{\frac{3}{9}}=\frac{\sqrt{3}}{3}$$

Norm of complex Vectors

$$\Vert v\Vert =\sqrt{v_1^{\ast }{v}_1+v_2^{\ast }{v}_2+\cdots +v_n^{\ast }{v}_n}$$

Proof: let $|z|=\sqrt{z^{\ast }z}\Rightarrow |z{|}^2=z^{\ast }z$

$$\therefore \Vert v{\Vert }^2=v_1^{\ast }{v}_1+v_2^{\ast }{v}_2+\cdots +v_n^{\ast }{v}_n=|v_1{|}^2+|v_2{|}^2+\cdots +|v_n{|}^2$$ $$v=(1+2i,i)\Rightarrow \Vert v{\Vert }^2=|1+2i{|}^2+|i{|}^2=(1+2^2)+1^2=6$$ $$\therefore \Vert v\Vert =\sqrt{6}$$

Dirac Notation of Norm Vector $\Rightarrow \Vert v\Vert =\Vert |v⟩\Vert =\Vert \text{k}v\Vert$