5.1 Basic notions

We will recall how geometric objects in a plane can be described using equations. These concepts are very useful because, as everyone knows, the output periphery of an overwhelming number of electronic devices are two-dimensional (monitors, paper, projectors, etc.).

Consider an orthogonal coordinate system with axes $x$, $y$ and origin $O$ in a plane. A point in this plane is described by two numbers called coordinates. For example, if a point $A$ has coordinates $(1,2)$, we write46 $A = (1,2)$, or, using square brackets, $A = [1,2]$. The point $A$ lies at the intersection of a line parallel to the $y$-axis which cuts the $x$-axis at number $1$ and a line parallel to the $x$-axis which cuts the $y$-axis at number $2$. The situation is shown in detail in Figure 5.1.

Figure 5.1: Orthogonal coordinate system and point $A = (1,2)$.

Another important geometric object is the vector. We will denote vectors by lower case letters with arrows, e.g. $\vec a,\, \vec b,\, \vec c$. We understand a vector as a pair of numbers47 giving a direction; if we have a vector $\vec a = (a_1,a_2)$ then the numbers $a_1$ and $a_2$ are called vector components of $\vec a$. We can add vectors and multiply them by a number using the rules

\begin{equation}\label{eq-vekt-operace}\tag{5.1} \alpha \cdot (a_1,a_2) := (\alpha a_1, \alpha a_2), \quad (a_1,a_2) + (b_1,b_2) := (a_1+b_1, a_2+b_2). \end{equation}

For obvious reasons we sometimes say that vector addition and scalar multiplication (multiplication of a vector by a number) defined in (5.1) are done „componentwise“. Equality of vectors is defined intuitively. We say that two vectors $\vec a = (a_1,a_2)$ and $\vec b = (b_1,b_2)$ are equal if their components are equal, i.e. if $a_1 = b_1$ and $a_2 = b_2$, and we write $\vec a = \vec b$. Geometric interpretation of vector addition and scalar multiplication is shown in Figure 5.2.

Figure 5.2: Geometric interpretation of scalar multiplication (a) and vector addition (b).

We can multiply a vector by a number. Can we also multiply two vectors? For that purpose we define scalar product 48. Standard49 scalar product of two vectors $\vec a = (a_1, a_2)$ and $\vec b = (b_1,b_2)$ is defined by this rule

\begin{equation*} \vec a \cdot \vec b := a_1 b_1 + a_2 b_2. \end{equation*}

The product is called scalar, because the result is not a vector but a number (a scalar). Furthemore, scalar product is related to the angle between vectors. The angle between two vectors $\vec a$ and $\vec b$ is $\alpha \in \langle 0,\pi)$, if and only if

\begin{equation*} \cos\alpha = \frac{\vec a \cdot \vec b}{\|\vec a\| \, \|\vec b \|}. \end{equation*}

Length of a vector $\vec a = (a_1, a_2)$ is defined by the Pythagoras' theorem. It is denoted by $\| \vec a \|$ and computed as

\begin{equation*} \| \vec a \| := \sqrt{a_1^2 + a_2^2} \quad \text{for} \ \vec a = (a_1,a_2). \end{equation*}

Note that the length can be also expressed using scalar product as $\| \vec a \| = \sqrt{\vec a \cdot \vec a}$.

You will study these and other geometric objects in the BIE-LIN course, for more than two dimensions as well.