Vectors
Notation
The set of all real numbers is denoted as $\mathbb{R}$ and the set of all vectors with $N$ coordinates, whose entries are real numbers, is denoted by $\mathbb{R}^N$.
Vectors are written as by $\boldsymbol{v}$ or $\vec{v}$.
The vector whose every entry is zero is called a zero vector.
But note that if $\boldsymbol{v} = \boldsymbol{0} \in \mathbb{R}^2$ and $\boldsymbol{u} = \boldsymbol{0} \in \mathbb{R}^3$, then $\boldsymbol{v} \neq \boldsymbol{u}$, nor can arithmetic operations, such as addition or subtraction be performed.
where $a_1, a_2, \ldots, a_k \in \mathbb{R}$, i.e. are scalars.
The result is a scalar.
The dot product can be generalised for more general vectors, called an inner product, written as $\langle \cdot, \cdot \rangle=c$, where $c$ is scalar.
The length of a vector is denoted by $\left| \vec{v} \right| = \sqrt{\vec{v} \cdot \vec{v}}$.
A unit vector has length one. A non-zero vector can be transformed into a unit vector by the transformation $\vec{v} \mapsto \vec{v} / \left| \vec{v} \right|$. This process is often called normalization, as the norm of a vector is often defined as $\sqrt{\langle \vec{v}, \vec{v} \rangle}$.
$$ \begin{equation*} \boldsymbol{a} \cdot \boldsymbol{b} = \left| \boldsymbol{a} \right| \left| \boldsymbol{b} \right| \cos\theta. \end{equation*} $$Thus, any two vectors, $\vec{v}$ and $\vec{u}$ are said to be orthogonal (or perpendicular) if $\vec{v} \cdot \vec{u}=0$.
$$ \begin{equation*} \cos\theta = \dfrac{\boldsymbol{a} \cdot \boldsymbol{b}}{\left| \boldsymbol{a} \right| \left| \boldsymbol{b} \right|}. \end{equation*} $$It can be derived from the Cauchy-Schwarz inequality.
For the Cauchy-Schwarz inequality, when the vectors $\vec{v}$ and $\vec{w}$ lie on the same line, then $\left| \vec{v} \cdot \vec{w} \right| = \left| \vec{v} \right| \left| \vec{w} \right|$.
For the triangle inequality, when the vectors point in the same direction, then the $\left| \vec{v} + \vec{w} \right| = \left| \vec{v} \right| + \left| \vec{w} \right|$.