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Definition of a vector

z

M z
M

. . . . . . . . .
0 M y y

M x
x

Figure 4.1 – Rectangular system of coordinates

4.2. Definition of a vector

There are quantities that are fully determined when assigning numeric values (length, area, vol-
ume, mass, temperature, etc) in certain sections of physics, mechanics and other technical dis-
ciplines. Such quantities are called as scalar. On the other hand, there are some quantities that
require, besides numeric values, assigning direction in space (force, speed, acceleration etc).
Such quantities are called as vector. Vector quantities are shown by means of vectors.

Definition 4.1. A directed interval in space, which has its own length is called a
vector (fig. 4.2).

B

−→
a
. . . .
A

Figure 4.2 – Vector in space

−→
A vector is marked as AB, where point A symbolises the beginning of the vector, point B
−→ −→
−→ −→
— its end. A vector can be marked with small Latin letters: a a , b , c , etc. The length of

−→
the vector is called its module: AB or | a | .
−→
When the beginning of the vector and its end are the same, such vector is called A the null-
vector and has the null length.
−→
− →
Vectors a , b located on the same straight line or on parallel lines are called colinear vectors.
A vector which has a unit length and the same direction with a given vector a is called its orth-
−→
−→
−→0
−→
vector and marked as a . Two vectors a , b are equal if they have the same direction and the
− →
−→
same length — it is written as a = b .
It follows from the definition that a vector can be self-removed to any point in space. Vectors
located in the same plane or in parallel planes are called complanar.
4.3. Linear operations with vectors

Linear operations include addition of vectors and multiplying of a vector by a number.

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