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$\newcommand{\la}{\langle}$ $\newcommand{\ra}{\rangle}$ $\newcommand{\vu}{\mathbf{u}}$ $\newcommand{\vv}{\mathbf{v}}$ $\newcommand{\vw}{\mathbf{w}}$ $\newcommand{\nc}{\newcommand}$ $\nc{\Cc}{\mathbb{C}}$ $\nc{\Rr}{\mathbb{R}}$ $\nc{\sp}[1]{\la #1 \ra}$ $\nc{\ol}[1]{\overline{#1} }$ $\nc{\norm}[1]{\left\| #1 \right\|}$

Geometry of Vectors III

Norm of a vector

The norm of a vector $\vv \in \Cc^n$, denoted by $\norm{\vv}$ is $\sqrt{\sp{\vv,\vv}}$.

In other words, $\sp{\vv,\vv} = \norm{\vv}^2$.

One should regard the norm of a vector as its length.

To get an idea, let's look at the simplest case when $n=1$.

Take a complex number $\alpha = a+bi$ ($a,b \in \Rr$), then $$ \norm{\alpha} = \sqrt{\sp{\alpha,\alpha}} = \sqrt{\alpha^*\alpha} = \sqrt{(a-bi)(a+bi)} = \sqrt{a^2 + b^2}. $$

which is the distance (Pythagorean Theorem) from the origin $(0,0)$ to $(a,b)$, i.e. the length of $\alpha$
(we identify the complex number $a+bi$ with the arrow from the origin to $(a,b)$).

For $\alpha \in \Cc$, its norm $\norm{\alpha}$ is also known as its modulus and also denoted by $|\alpha|$.

Exercise. Check that for $\alpha \in \Cc$ and $\vv \in \Cc^n$, $\norm{\alpha \vv} = |\alpha|\norm{\vv}$ .

Ans $\norm{\alpha \vv}^2 = \sp{\alpha\vv,\alpha\vv} = (\alpha\vv)^*(\alpha\vv) = \vv^* \alpha^* \alpha \vv = \vv^* \ |\alpha|^2\vv = |\alpha|^2 \vv^* \vv = |\alpha|^2\norm{\vv}^2$.

Since norms of vectors are nonnegative, the result follows by taking square root on both sides.

z = 2+I; arrow2d((0,0),(z.real(),z.imag()))
#the arrow from the origin to the point (a,b) represent the complex number a+bi.

Unit vectors

A unit vector is a vector of length $1$.

For any $\vv \neq \mathbf{0}$, $\norm{\vv} > 0$. So $|\norm{\vv}| = \norm{\vv}$. By applying the exercise to $\alpha = 1/\norm{\vv}$, we get

$$ \norm{\dfrac{\vv}{\norm{\vv}}} = \left|\dfrac{1}{\norm{\vv}}\right|\norm{\vv} = \dfrac{1}{\norm{\vv}}\norm{\vv} = 1. $$

Thus, $\vv/\norm{\vv}$ is a unit vector.

We call $\vv/\norm{\vv}$ (resp. $-\vv/\norm{\vv}$) the unit vector in the direction (resp. in the opposite direction) of $\vv$.

In general, unit vectors should be treated as directions so we also refer $\hat{\vv} := \vv/\norm{\vv}$ simply as the direction of $\vv$.

The zero vector has no direction.

Two nonzero vectors $\vv$ and $\vw$ are in the same direction (resp. in opposite direction) if $\hat{\vv} = \hat{\vw}$ (resp. $\hat{\vv} = -\hat{\vw}$).

Parallel

Two nonzero vectors $\vv, \vw$ are parallel if they are nonzero multiple of one another, i.e. $\vw = \lambda \vv$ for some $\lambda \neq 0$ (then $\vv = (1/\lambda)\vw$).

By convention, we declare the zero vector to be parallel to every vector.