If three pairwise perpendicular lines are drawn through a point in space, on each of which a direction is chosen and the unit of measurement of the segments is chosen, then they say a rectangular coordinate system in space is specified.
Straight lines with directions selected on them are called. axes coordinates.(Ooh, Ooh, Oz)
Planes passing through the coordinate axes are called coordinate planes. (Oxy, Oyz, Ozx)
The transformation of figure F to figure F1 is called
movement, if it maintains the distance between
points, i.e. translates any two points A and B of the figure
F to points A1 and B1 of figure F1 so that AB=A1B1.
Symmetry relative to the plane
Let A– arbitrary fixed plane. From
X points lower a perpendicular to the plane A (ABOUT –
the point of intersection of it with the plane A) and on it
continuation beyond the point ABOUT segment OX1 is postponed,
equal to OX. Points X and X1 are called symmetrical
relative to the plane A.
Transformation of figure F to F1, in which each point X
figure F goes to point X1, symmetrical to X relative
plane A, called symmetry transformation
relative to the plane A. In this case, the figures F and F1 are called
symmetrical relative to the plane A.
Figure 1 shows two spheres,
symmetrical relative to the plane A.
If the symmetry transformation
translates relative to the plane
figure into itself, then the figure is called
symmetrical about
plane A, and the plane A called
plane of symmetry.
Figure 2 shows two planes
symmetry of the sphere. Note that the sphere
there are an infinite number of such planes of symmetry
a bunch of. The cube also has planes
symmetry. Figure 3 shows two of them.
Parallel transport in space is called
such a transformation under which an arbitrary point
(x; y; z) of figure F goes to point (x+a; y+b; z+c), where a,
b, c are constants. Parallel transport in space is given
formulas x1=x+a, y1=y+b, z1=z+c. In Figure 4, prism ABCA1B1C1
during parallel transfer it goes into the prism A’B’C’A’ 1B’ 1C’ 1.
Let us formulate some properties of parallel transfer:
Parallel transfer is movement.
With parallel translation, points are shifted along parallel (or coinciding) lines by the same distance.
With parallel translation, a straight line goes into a parallel line (or into itself).
Whatever the two points A and A1, there is, and moreover, a unique, parallel transfer, in which point A goes to point A1.
With parallel translation in space, each plane transforms either into itself or into a plane parallel to it.
Let F be a given function and O be a fixed point (Fig. 5).
Let us draw a ray OX through an arbitrary point X of the figure F and
Let us plot on it the segment OX1 equal to kOX, where k –
positive number. Transformation of figure F, in which
each of its points X goes to point X1, constructed
in this way is called homothety relatively
center O. The number k is called homothety coefficient.
Figures F and F1 are called homothetic.
Converting a figure to a figure is called
similarity transformation, if at the same time
converting the distance between points
change (increase or decrease)
the same number of times. This means that if
arbitrary points A and B of figure F in this case
transformation go to points A1 and B1
figure F1, then A1B1=kAB, where k >0.
The number k is called similarity coefficient. At
k=1 similarity transformation is movement.
Angle between planes α and β,
Angle between planes α and β,
which intersect along
straight line c is called angle
between the lines along which
third plane γ,
perpendicular to the line
intersections, crosses
planes α and β.
Angle between parallel
planes is considered equal to 00.
The angle between the planes is not
exceeds 900.
On the straight line from the intersection of planes α and β, select point C; through C planes
α and β draw straight lines a and b perpendicular to c. The angle between lines a and b is equal to the angle between the planes.
A; A belongs With β : AB perpendicular With; AA1 perpendicular β A And β a-priory.
2. Take point A belongs to A; A belongs With; let us drop perpendiculars from it onto the straight line c and the plane β : AB perpendicular With; AA1 perpendicular β . Let's connect points B and A1: A1B perpendicular to c using the theorem of three perpendiculars; angle ABA1 – angle between planes A And β a-priory.
The angle between a straight line and the plane that contains it
intersects is called the angle between this line and its
projection onto a plane.
To construct the projection of straight line a onto a plane, it is enough
find two surface points: for example, the point of intersection
straight line a and the plane and base of any perpendicular,
lowered from the second point of line a onto the plane.
The angle between parallel straight lines and plane α is considered equal to 00
The angle between a perpendicular line and a plane is 900.
Angle between
Angle between
interbreeding
an angle is called a straight line
between the lines that
intersect and
parallel to data
crossing straight.
If the angle between
interbreeding
straight lines equal 900, then they
are called
perpendicular.
The orthogonal projection of a point onto a plane is the base of a perpendicular drawn from a given point onto the plane
The projection of a segment onto a plane is a segment connecting the projections of its ends.
The projection of a polygon onto a plane is a figure bounded by the projections of the sides of the polygon onto this plane.
The area of the orthogonal projection of a polygon is equal to the product of its area and the cosine of the angle between the plane of the polygon and the projection plane.
State educational institution
primary vocational education
« Professional institute No. 5" Belgorod
Lesson summary
in mathematics on the topic:
Rectangular coordinate system in space
for 11th grade students
Prepared by:
Kobzeva Irina Alekseevna,
teacher of computer science and mathematics
GOU NPO PU No. 5
Belgorod 2010
Lesson topic : Rectangular coordinate system in space. Vector coordinates
Lesson objectives: - develop logical and spatial thinking
Introduce the concept of a coordinate system in space, vector coordinates
Literature: Geometry 10-11 grade L. S. Atanasyan, M.: Education, 2006
During the classes:
Org. Moment
Announcing the topic and purpose of the lesson.
Explanation of new material
Rectangular coordinate system in space.
If three pairwise perpendicular lines are drawn through a point in space, a direction is selected on each of them (it is indicated by an arrow) and a unit of measurement for the segments is selected, then it is said that rectangular system coordinates in space (Fig. 121). Straight lines with directions chosen on them are called coordinate axes, and their common point is origin. It is usually denoted by the letter O. The coordinate axes are designated as follows: Ox, Oy, O z - and have names: abscissa axis, ordinate axis, applicate axis. The entire coordinate system is designated Oxyz. The planes passing through the coordinate axes Ox and Oy, Oy and Oz, Oz and Ox, respectively, are called coordinate planes and are designated Oxy, Oyz, Ozx.
Point O divides each of the coordinate axes into two rays. A ray whose direction coincides with the direction of the axis is called positive semi-axis, and the other beam negative semi-axis.
IN 
In the rectangular coordinate system, each point M of space is associated with a triple of numbers, which are called its coordinates. They are determined similarly to the coordinates of points on the plane. Let us draw three planes through point M, perpendicular to the coordinate axes, and denote by M 1, M 2 and M 3 the intersection points of these planes, respectively, with the abscissa, ordinate and applicate axes (Fig. 122). The first coordinate of point M (it is called abscissa and is usually denoted by the letter x) is defined as follows: x = OM 1, if M 1 is the point of the positive semi-axis; x = - OM 1, if M 1 is the point of the negative semi-axis; x = 0 if M 1 coincides with point O. Similarly, using point M 2 the second coordinate ( ordinate)
y point M, and using point M 3 the third coordinate ( applicate) z point M. The coordinates of point M are written in parentheses after the designation of the point: M (x; y; z), with the abscissa indicated first, the ordinate indicated second, and the applicate third indicated. Figure 123 shows six points A (9; 5; 10), B (4; -3; 6), C (9; 0; 0), E (4; 0; 5), E (0; 3; 0 ), F (0; 0; -3).
E 
if point M (x; y; z) lies on the coordinate plane or on the coordinate axis, then some of its coordinates are equal to zero. So, if M € Oxy, then the applicate of the point M is equal to zero: z = 0. Similarly, if M with Oxz, then y = 0, and if M € Oyz, then x = 0. If M € Ox, then the ordinate and applicate of the point M are equal to zero: y = 0 and z = 0 (for example, at point C in Figure 123). If M € Oy, then x = 0 and z = 0; if M € Oz, then x = 0 and y = 0. All three coordinates of the origin are equal to zero: 0 (0; 0; 0).
Vector coordinates
Z 
let us define a rectangular coordinate system Oxy in space z. On each of the positive semi-axes we plot from the origin of coordinates unit vector, i.e. a vector whose length is equal to one. Let us denote by i unit vector of the x-axis, through j- unit vector of the ordinate axis and through k unit vector of the applicate axis (Fig. 124). Let's call vectors i, j, k coordinate vectors. Obviously, these vectors are not coplanar. That's why any vectoraand can be expanded into coordinate vectors, i.e., represented in the form
and the expansion coefficients x, y, z are determined in a unique way.
TO 
coefficients x, y and z in the expansion of vector a in coordinate vectors are called vector coordinatesain this coordinate system. We will write the coordinates of the vector a in curly brackets after the vector designation: a (x; y; z). Figure 125 shows a rectangular parallelepiped having the following dimensions: OA 1 = 2, OA 2 = 2, OA 3 =4. The coordinates of the vectors shown in this figure are: a (2; 2; 4), b(2; 2; -1), A 3 A (2; 2; 0), i(1; 0; 0), j (0; 1; 0), k(0; 0; 1).
Since the zero vector can be represented as 0 = o i+ оj+ 0k then all coordinates of the zero vector are equal to zero. Further, the coordinates of equal vectors are respectively equal, i.e. if the vectors a(x 1, y 1, z 1) and b(x 2, y 2, z 2) are equal, then x 1 = x 2, y 1 = y 2 and z 1 = z 2 (explain why).
Let's consider rules, which allow using the coordinates of these vectors to find the coordinates of their sum and difference, as well as the coordinates of the product of a given vector by a given number.
1 0 . Each coordinate of the sum of two or more vectors is equal to the sum of the corresponding coordinates of these vectors. In other words, if a (x 1, y 1, z 1) and b(x 2, y 2, z 2) are these vectors, then the vector a + b has coordinates (x 1 + x 2, y 1 + y 2, z 1 + z 2 ).
2 0 . Each coordinate of the difference of two vectors is equal to the difference of the corresponding coordinates of these vectors. In other words, if a (x 1, y 1, z 1) and b(x 2 y 2; z 2) are these vectors, then the vector a- b has coordinates (x 1 - x 2, y 1 – y 2, z 1 - z 2).
3 ABOUT. Each coordinate of the product of a vector and a number is equal to the product of the corresponding coordinate of the vector and this number. In other words, if a (x; y; x) is a given vector, α is a given number, then the vector α
a has coordinates (αх; αу; αz).
Any vector ā can be expanded into coordinate vectors, i.e. represented in the form: The zero vector can be represented in the form: The coordinates of equal vectors are respectively equal, i.e., if ā ( x 1 ; y 1 ; z 1 ) = b ( x 2 ; y 2 ; z 2 ), then x 1 = x 2, y 1 = y 2, z 1 = z 2.
A vector whose end coincides with a given point, and whose beginning coincides with the origin of coordinates, is called the radius vector of a given point. The coordinates of any point are equal to the corresponding coordinates of its radius vector. M (x; y; z) OM (x; y; z) A (x 1 ; y 1 ; z 1), B (x 2 ; y 2 ; z 2) AB (x 2 – x 1 ; y 2 – y 1 ; z 2 – z 1)
