I. Vectors, straight lines, circles

Point

The lenght of a line segment

Coordinates of the center of aline segment

Vector

Vector – origin at point A and the end at point B

Zero vector

An opposite vector with respect to vector v

The lenght of a vector

The sum of two vectors,

wvu

The product of a number anda vector vau

The scalar product of vectors,

wva

The vectorial product,wvu

Geometrical scalar and vectorial products:

Equations of a straight line:

■■

■

■■

■■

General

Segmental

Directional

Determinant

Parametrical

Two - points

B

An Vector perpendicular to the straight line

q

pv

Vector parallel to the straight line

δ, the distance betweenpoint P(x0,y0) and the straight line

φ, the angle between two straight lines

The condition of perpendicularity

The condition of parallelity

1. Write an equation of a straight line, perpendicular to the line x + 3y – 7 = 0, passing through the point A=(a;b), where a and b are roots of the equation:

52594 22 xx

2. Draw a line through the point A=(a;b), which is perpendicular to the line x + 2y – 5 = 0, where a is the solution of the equation:

41

16

81

9

4

xx

and b is the solution of the equation:

01logloglog 238 x

3. There are two straight lines k and l of equations k : 3x – y = –18 l : x + y = 2 and the point A=(3;–1). Find such a point P on OX axis, so that vectors are perpendicular, giving that point B is a common point of lines k and l.

ABandAP

4. Points A=(1;2); B=(–1;–1); C=(5;2) are vertexes of a triangle. a) Write an equation of a straight line which contains the triangle’s height from the vertex A. b) Determinate point’s D coordinates, so that the tetragon ABCD is a parallelogram.

SET OF PROBLEMS

1. Take into account the following sets:

13:

11

13:

110log:

6

2

2

2

xxRxC

x

xxRxB

xRxA

Determine: CBA

2. The third term of an arithmetic sequence is equal to 3log8 3

the sixth term is equal to 3732

35

Determine this sequence.

How many initial terms one needs to take, so that their sum is 14650

3. It is given the polynomial

cxbxaxxW 23

The roots of the polynomial are numbers: –1; p; q.Determine coefficients a; b; c of W (x), if p is the solution of the equation

2264 22 xxx and 2

23

1

123lim

nn

nnq

n

4. The straight line of the equation y + 3x + 2 = 0 intersects the parabola

822 xxy in points A and B.

a) Calculate the area and the circumference of the triangle ABS, where S is the vertex of the parabola.b) Write down a circle’s equation which is circumscribed on this triangle.