*Vectors* are lines which have a constant direction and lenght and it can it move in given system of coordinators.
Coordinates of vector *u* are (*u1,u2*) in surface and (u1,u2,u3)
in space.

A vectors are usually marked the same way as the coordinates of a points
*v*
= (*x1,y1*), where
*x1*
is the *x*-axis vector coefficient and
*y1*
is its
*y *coefficient. This vector can be drawn as a line from the point A
with coordinates (*a0,a1*)
to B dot with coordinates (*b0,b1*).

*x *unit vector is *i* = (*1, 0*)
and *y* unit is *i* = (*1, 0*). This is valid in a surface,
in space z coordinate is to be added.
Unit vectors then will be (*1,
0, 0*), (*0, 1, 0*) a (*0, 0, 1*).
(vectors that have a length of one)

Vector addition is performed by adding the terms together and their results are also vectors

**Example** :

Vectors are added and substraction of vectors *u* = (*u0,
u1, u2*) and
*v* =(*v0,
v1, v2*) :

*w* = *u* + *v *= (*u0 + v0,
u1 + v1, u2 +
v2*)
*w* = *u* - *v* = (*u0 - v0,
u1 - v1, u2 -
v2*)

Lenght of vectors *AB=(x,y,z)* represents vector
absolute value and it is calculated as follows:

|*AB*| = sqrt(*x^2 + y^2 + z^2*).

It is the same as the distance between dots (*x,y,z*) and (*0,0,0*).

A matrix is a rectangular array of numbers. Its elements are doubly indexed , and by convention the first inex indicates the row and the second indicates the column. Specially selected table of real and complex numbers is called a matrix.

Number *a(i,j)* is an element of matrix that contains*n*
columns and *m* lines, for which it is valid that 1 ≤ *i* ≤ *m*
and 1 ≤ *j* ≤ *n*.
The size of the matrix *m
x n*, which is inscribed as**A**(*m
x
n*). The first index of *i* element of the matrix *a(i,j)*
shows in which line the element is situated, analogically of the index *j* shows the column.
For example a(2,4) refers to the element in the second row, fourth column.

Let`s see now a matrix type *1 x
n*. This matrix can be called a horizontal or line vector.
It is marked by lower-case letters :

a= | a(1) a(2) . . . a(n) |

A matrix is called a square one if the number of lines and columns in the matrix is the same (e.g. matrix 3x3, 4x4).

Matrix is a diagonal one if all elements above and under the diagonalare zero.
*A diagonal* is a group of dots of matrix with the same index in
x and y directions.

| 1 0 0 | Diagonal`s matrixA= | 0 1 0 | | 0 0 2 |

A special case of diagonal matrix is a unit matrix
(or identity matrix), whose all elemnts are 1.
It is usually marked by symbol **I**.

| 1 0 0 0 | | 0 1 0 0 | A unit matrixI= | 0 0 1 0 | | 0 0 0 1 |

A matrix with all its elements being zero, is a *zero matrix*.
It is marked **O**.

Matrix **B** is *inverse* to matrix **A**
if it is valid that:
**A** x **B** = **I**, alebo **B** = **A** ^
(-1)

Matrix adding is defined for matrixes of the same type in the way that
element a(i,j) of **A** matrix is added to each element b(i,j) of
**B** matrix **B, **for 1 ,...,*m*
and 1 ,...,*n*.

**Example** : Matrix adding 3x3.

| 0 2 0 | | 1 0 0 | | 1 0 0 | A = | 7 5 1 |, B = | 2 1 0 |, A+B = | 9 6 1 |. | 2 4 5 | | 3 4 1 | | 5 8 6 |

By a *scalar* we understand one number from defining group.
When multiplying, every element of the matrix is by the scalar.

**Example**:

| 1 2 3 | A = | 0 1 -2 |, k = 3 |-1 0 2 | | 1 2 3 | | 3 6 9 | A*k = | 0 1 -2 | * 3 = | 0 3 -6 | |-1 0 2 | |-3 0 6 |

To simplify calculation of the transformation we use representation of dots
by the means of *homogeneous coordinates*.

Four organized numbers [*x, y, z, w*] is called a right-angle
homogeneous coordinates of point *P* with carthezian coordinates
[X, Y, Z] in three-dimensional space, if

x y
z

*X* = ---
, *Y* = --- , *Z *= --- , *w * =
R - {0}.

*w* *w*
*w*

Most frequently it is selected *w* = 1, because it is easy.
In such a case, homogene coordinates of the point are [X, Y, Z, 1].

Matrix 4x4 reprezenting a linear transformation of point *P* = [*x, z, y, w*]
onto point *P'* = [*x', y'
, z' , w'*] will be marked **A**.

| a11 a12 a13 0 |A= | a21 a22 a23 0 | | a31 a32 a33 0 | | a41 a42 a43 1 |

The value of the point *P'* from the point
*P* will be calculated as follows:

| a11 a12 a13 0 |
*P'* = [*x' y' z' w'*] = *P****A** =
[*x y z w*] * | a21 a22 a23 0 |.

| a31 a32 a33 0 |

| a41 a42 a43 1 |

For matrix 3x3 we proceed similarly.

Copyright (c) 1999-2018 Juraj Štugel. www.netgraphics.sk