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. a,b,c. We understand a vector as a pair of numbers47 giving a direction; if we have a vector a=(a1,a2) then the numbers a1 and a2 are called vector components of a. We can add vectors and multiply them by a number using the rules

(5.1)α(a1,a2):=(αa1,αa2),(a1,a2)+(b1,b2):=(a1+b1,a2+b2).

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 a=(a1,a2) and b=(b1,b2) are equal if their components are equal, i.e. if a1=b1 and a2=b2, and we write a=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 a=(a1,a2) and b=(b1,b2) is defined by this rule

ab:=a1b1+a2b2.

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 a and b is α0,π), if and only if

cosα=abab.

Length of a vector a=(a1,a2) is defined by the Pythagoras' theorem. It is denoted by a and computed as

a:=a12+a22for a=(a1,a2).

Note that the length can be also expressed using scalar product as a=aa.

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