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 containsn columns and m lines, for which it is valid that 1 i m and 1 jn. The size of the matrix m x n, which is inscribed asA(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 matrix A = | 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 matrix I = | 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.