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Vectors. The scalar, vector and mixed product of vectors. The equation of the line in the plane. The curves of the second order

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Theoretical questions: 1. How Rectangular Cartesian coordinates are given on the plane?

2. What is connection between polar and Rectangular coordinates of points?

3. What formula finding of distance between two points?

4. Equation of straight line with angular coefficients.

5. General equation of straight line.

6. Equation of straight line with given angular coefficients and passing through a given point.

7. Equation of straight line passing through given two points.

8. Equation of straight line in segments.

9. Angle between two straight lines.

10. The parallelism conditions of two straight lines.

11. The perpendicularity conditions of two straight lines.

12. The distance from point to straight line.

15. Which of the following statements are true and which are false for all lines and planes in xyz -space: (a) Two lines parallel to a third line are parallel. (b) Two lines perpendicular to a third line are parallel. (c) Two planes parallel to a third plane are parallel. (d) Two planes perpendicular to a third plane are parallel. (e) Two lines parallel to a plane are parallel. (f) Two lines perpendicular to a plane are parallel. (g) Two planes parallel to a line are parallel. (h) Two planes perpendicular to a line are parallel. (i) Two lines are either parallel or intersect. (j) Two planes either are parallel or intersect. (k) A line and a plane either are parallel or intersect. (l) Two non-parallel lines either intersect or there are parallel planes that contain them.

 

Classroom assignments:

  1. Define the distance between points А(3;8) and В(-5;14).
  2. On axe Ох to find a point with the distance 13 unit from point М(2;5).
  3. Show that points А(6;3), В(1;-2), С(-2;-5) belong to one line.
  4. The segment bounded by points А(3; -2) and В(6;4), was divided on three equals parts.
  5. Define coordinates of vertices triangle, if it is known the middle its sides Р(2;3), Q(5;4), R(6;-3).
  6. Find polar coordinates of points , , , if the pole coincide with point of origin and polar axe -- with position direction of abscissa axe.
  7. Find Rectangular coordinates of points: .
  8. Write equation of line passing through the points: а) А(0;2), В(-3;7); б) А(2;1), В(4;1).
  9. Find angular coefficient to line and ordinate of point its intersection with axe Оу, if it is known line pass through points А(1;1) and В(-2;3).

10. The coordinates of points М1(-3;5) and М2(4;6) are given and equation of line d .

To demand:

a). construct line d and points М1 and М2;

b). calculate the distance from point М1 to line d

c). write equation of line passing through point М1, parallel to line d

d). write equation of line passing through point М1, perpendicular to line d

e). write equation of line М1М2;

f). clarify mutual disposition of lines М1М2 and d;if they are not parallel, to define tangent of angle between them and to find coordinate of points its intersection.

11. Give equations of (a) the line through (5, 6, 7) and parallel to the line x = 4, y = 6 – t, z = 9 + 2 t, (b) the line through the origin and perpendicular to the plane 3 x – 4 y + 5 z – 18 = 0

12. What are the numbers a and b if is perpendicular to the plane 4 x –2 y + z = 5?

13. Find the component form of the vector:

a. The vector where .

b. The vector where O is the origin and is the midpoint of segment , where .

c. The vector from the point to the origin.

d. The sum of and , where .

14. Find a) the direction of and b) the midpoint of line segment .

a. and

b. and

 


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