1. Definition

Geometrically, a vector is an arrow having the three following characteristics :

  • a direction (the line that contains it)
  • a sense (top, right, bottom, left)
  • a length (also called norm)

In Figure 5.1, we have the following :

  • a et c are equal. a=c. The coordinates are : c=(32).
  • a et b have the same direction, the same length but an opposite sense. a=b.
  • d et e have the same direction, the same sense, but a different length. d=2e.
vectors in plane

Figure 5.1 : vectors in R2

2. Addition

The addition of two vectors is defined as follows :

a+b=(a1an)+(b1bn)=(a1+b1an+bn)=(b1+a1bn+an)=b+a

Geometrically, the sum of two vectors is obtained by putting them end to end, as showed in Figure 5.2. This is the so-called parallelogram law.

Example I

vector addition

Figure 5.2 : vector addition in R2

3. Subtraction

The vector subtraction is nothing but adding a vector whose sense is reversed.

Example II

subtraction of vectors

Figure 5.3 : vector subtraction in R2

4. Multiplication by a scalar (number)

A vector multiplied by a scalar gives another vector. It is is defined as follows :

Example III

vector multiplication by scalar

Figure 5.4 : multiplication by a scalar in R2

Recapitulation

Geometrically, a vector is an arrow having a direction, a sense and a length. Two vectors are equal if they have those same characteristics or, in other words, if they have the same coordinates.

Geometrically, a vector can been seen as a point. The concept of vector generalizes the concept of point.

If two vectors have the same direction, then they are multiples of one another, like d and e in Figure 5.4. In that case, they are said to be collinear or, more generally, linearly dependant.

More generally, a vector is an algebraic and abstract concept which is beyond the scope of the present course.