Euclidean Vector

Definition

Geometric object that has magnitude and direction

Elements of a Vector Space

Properties

Length

Vector

Real

Let $\mathbf{x}\in {\mathbb{R}}^n$ be an $n$-dimensional vector, then the $L_p$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}{\Vert }_p:={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{1/p}$

Complex

Let $\mathbf{x}\in {\mathbb{C}}^n$ be an $n$-dimensional complex vector, then the $L_1$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}\Vert ={\mathbf{x}}^{†}\mathbf{x}=\sqrt{{\sum }_{i=1}^n{\bar{x}}_i{x}_i}$

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Addition and Subtraction

for $\mathbf{x}\in {\mathbb{R}}^n,\mathbf{y}\in {\mathbb{R}}^n$ $\mathbf{x}\pm \mathbf{y}=[x_1\pm y_2,x_2\pm y_2,\dots ,x_n\pm y_n{]}^{⊺}$

Distance

for $\mathbf{x}\in {\mathbb{R}}^n,\mathbf{y}\in {\mathbb{R}}^n$ $||\mathbf{x}-\mathbf{y}||=||\mathbf{y}-\mathbf{x}||=\sqrt{{\sum }_{i=1}^n(x_i-y_i{)}^2}$

Scalar Multiplication

for $\mathbf{x}\in {\mathbb{R}}^n$ $k\mathbf{x}=[k{x}_1,k{x}_2,\dots ,k{x}_n{]}^{⊺}$

Inner Product

Vector

Real

for $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$ $⟨\mathbf{x},\mathbf{y}⟩=||\mathbf{x}|\cdot ||\mathbf{y}||\cos \theta ={\sum }_{i=1}^n{x}_i{y}_i$ where $||\cdot ||$ is a Norm

Complex

for $\mathbf{x},\mathbf{y}\in {\mathbb{C}}^n$ $⟨\mathbf{x},\mathbf{y}⟩=\mathbf{x}\overline{\mathbf{y}}={\sum }_{i=1}^n{x}_i{\bar{y}}_i$

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Vector Product

Cross Product

Definition

$\begin{array}{rl}\mathbf{v}\times \mathbf{w}& =||\mathbf{v}||||\mathbf{w}||\sin (\theta )\mathbf{n}=\left(\left|\begin{array}{cc}{v}_2& v_3\\ w_2& w_3\end{array}\right|,-\left|\begin{array}{cc}{v}_1& v_3\\ w_1& w_3\end{array}\right|,\left|\begin{array}{cc}{v}_1& v_2\\ w_1& w_2\end{array}\right|\right)=[(v_2{w}_3-v_3{w}_2),(v_3{w}_1-v_1{w}_3),(v_1{w}_2-v_2{w}_1)]\\ & =\det \left(\left[\begin{array}{ccc}i& v_1& w_1\\ j& v_2& w_2\\ k& v_3& w_3\end{array}\right]\right)\end{array}$ where the $\mathbf{v},\mathbf{w}\in {\mathbb{R}}^3$, the $\theta$ is the angle between $\mathbf{v}$ and $\mathbf{w}$, and the $\mathbf{n}$ is a unit vector perpendicular to the both $\mathbf{v}$ and $\mathbf{w}$.

Facts

$\mathbf{u}\times \mathbf{v}=-(\mathbf{v}\times \mathbf{u})$

$\mathbf{u}\times (\mathbf{v}+\mathbf{w})=(\mathbf{u}\times \mathbf{v})+(\mathbf{u}\times \mathbf{w})$ $\mathbf{u}\times (\mathbf{v}-\mathbf{w})=(\mathbf{u}\times \mathbf{v})-(\mathbf{u}\times \mathbf{w})$

$(\mathbf{u}+\mathbf{v})\times \mathbf{w}=(\mathbf{u}\times \mathbf{w})+(\mathbf{v}\times \mathbf{w})$

$k(\mathbf{u}\times \mathbf{v})=(k\mathbf{u})\times \mathbf{v}=\mathbf{u}\times (k\mathbf{v})$

$\mathbf{u}\times 0=0\times \mathbf{u}=0$

$\mathbf{u}\times \mathbf{u}=0$

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