Lecture 1. Rectangular Cartesian coordinate system
In this lecture, we are going to discuss the concept of an orthonormal basis and a Cartesian coordinate system. First, we are going to recall a number of concepts and facts that we have studied in previous lectures. We have already worked with the concept of a geometric vector.
A vector is an object that is defined by its length and direction. From any point, you can set a vector equal to some given vector. Е2 denotes a set of collinear vectors, that is, parallel to the same plane. Е3 is the set of all vectors of a geometric space. Е2 is a two-dimensional vector space, and Е3 is a three-dimensional vector space.
Two or more vectors are linearly dependent if there is a vector that is a linear combination of the other vectors of the given system. A system consisting of a single zero vector is linearly dependent. If the vectors are not linearly dependent, they are linearly independent.
The basis of a vector system is any of its linearly independent subsystems, through which each vector of the original system is expressed.
In the Е2 space, any two non-collinear vectors form a basis. If we take two vectors a and b, then any vector from a given plane can be represented as a linear combination of vectors a and b. Let's present it in the drawing. If the vector v is noncollinear to the vectors a and b, then it is the diagonal of a parallelogram whose sides are on straight lines containing vectors a and b.
The same fact holds for the space Е3. First, Е3 is a three-dimensional space. Any three non-planar vectors form its basis. We take the non-planar vectors a, b, and c. Then we choose an arbitrary vector v and represent it as a linear combination of the vectors a, b, and c. Let's show it in the drawing. Note that if the vector v is not coplanar to any two of the original vectors, then it is the diagonal of a parallelepiped whose sides are on the straight lines containing the vectors a, b, and c.
In any vector space, each vector is uniquely decomposed by the basis. In other words, if we fix the vectors in the basis, the expansion coefficients are determined unambiguously. These coefficients are the coordinates of the vector. In the Е2 space, vectors are defined by two coordinates, which are its expansion coefficients by the chosen basis. In a three-dimensional space, three coordinates define any vector.
We need to consider an orthonormal basis. Let's take a two-dimensional space Е2 and choose two vectors with an angle between them equal to 90˚. These vectors are orthogonal or perpendicular. The lengths of these vectors are equal to 1. Thus, we get the basis, which is an orthonormal basis on the plane.
Similarly, we can define a basis in the Е3 space. We take three vectors. The angle between any two of them is 90˚. We take pairwise orthogonal vectors with lengths equal to 1. We get an orthonormal basis in the Е3 space.
We can determine the coordinates of an arbitrary vector in this basis.
In order to determine the coordinates of a point, we need to introduce a coordinate system. To do this, we choose an orthonormal basis. First, we are going to consider the plane, that is, consider the basis of two vectors i and j. Let's set these vectors from one point. Let's denote the point with O. Let's call it the coordinate origin. Each of the basis vectors sets its own axis. The axis directed along the first vector i is the abscissa axis. The axis directed along the second vector j is the ordinate axis. Let us take an arbitrary point. For this point, we can consider a vector with a start at the coordinate origin and an end at this point. Thus, we get the vector OA, which is the radius vector of this point. Let's draw the perpendiculars on the coordinate axis from the point A and get the points Аx and Аy, which are projections of the point A. Аx is the projection of the point A on the abscissa axis, and Аy is the projection on the ordinate axis. Each of these points sets a number, namely, the point Аx sets the number а1, which we calculate according to the rule specified on the slide. а1 is either equal to the length of the vector x, or it has the opposite value. It all depends on whether the given vector is co-directed or opposite to the basis vector. Similarly, the point Аy, the second projection, determines the number. These numbers are projections of the OA vector on the coordinate axis. We can calculate these projections using the formulas. For example, the number а1 is calculated by the specified equation, where the angle is under the cosine. This angle is chosen between the positive direction of the abscissa axis and the given vector.
The example is shown in the figure. We have an obtuse angle, so the cosine of this angle is less than 0, which means that the number а1 is negative. We can find the second number using a similar formula. To do this, we are going to consider another angle between the vector and the ordinate axis, the angle b. For this example, the angle b is acute, so the cosine is greater than 0, and therefore the number а2 is positive.
The numbers а1 and а2 are projections of the vector a on the coordinate axis. The point Аx defines the vector ОАx, and the point Аy defines the vector ОАy. These vectors are also projections of the OA vector on the coordinate axis, with the OA vector being the sum of its projections. Each projection is a vector collinear to the basis one. The vector a can be represented as a linear combination of basis vectors, and the coefficient r and s in the expansion are equal to the projections а1 and а2.
We can conclude that the rectangular coordinates of the vector a (r, s) coincide with the projections of this vector on the coordinate axis.
Coordinates of a point are the coordinates of its radius vector.
Similarly, we can consider the coordinate system in the Е3 space. We take an orthonormal basis made up of vectors i, j, and k. We fix the point O; it is the coordinate origin, and we setthese vectors from it. We get a coordinate system that is set by the axes. We get the third axis, which is the applicate axis. Each two axes define the coordinate planes Оxy, Оxz and Oyz.
Let us consider an arbitrary point. It defines its radius vector. We denote the point as A and get the vector OA.
Let's define the concept of projection. To do this, we draw the plane through the point A, perpendicular to the abscissa axis. This plane intersects the axis at the point Аx. This point is the projection of the vector OA on the axis Ox. Similarly, we define the other two projections (they are the point Аy on the Oy axis and the point Аz on the Oz axis). These projections define three numbers a1, a2, and a3, which can be calculated using similar formulas via the cosine.
The resulting vectors ОАx, ОАy and ОАz are also projections of the vector on the coordinate axis. The source vector is the sum of its projections. The coefficients of this sum, standing for the vectors i, j, and k are the coordinates of this vector. Thus, these coordinates are equal to the projections a1, a2, and a3. Thus, just as on the plane, the coordinates of the vector coincide with its projections on the coordinate axis for a three-dimensional space. The coordinates of a point are the coordinates of its radius vector.
We have defined two coordinate systems on the plane and in space. These coordinate systems are rectangular. The term "rectangular" is justified by the fact that we consider an orthonormal basis. In this basis, the angle between any two vectors is 90 degrees. These coordinate systems are also called Cartesian in honor of Rene Descartes, a 17th-century mathematician who is considered one of the founders of analytical geometry. Therefore, certain coordinate systems are called rectangular Cartesian coordinate systems.
Let's consider some theorems.
First, it is known that in any space, when adding vectors, their coordinates are added, which means that a similar property takes place in the geometric spaces Е2 and Е3. That is, when adding vectors, we add up their coordinates. Similarly, if we multiply a vector by a number, then we multiply its coordinates by that number.
Second, let us have a MN vector. We know the coordinates of its starting and ending points M and N. In this case, we can find the coordinates of the vector MN using the following rule: it is necessary to subtract the coordinates of the starting point from the coordinates of the ending point. Let's try to illustrate this property. We are going to consider the coordinate system on the plane. We take a vector set by two points M and N. Let us consider their radius vectors OM and ON. The coordinates of a point are the coordinates of its radius vector. In this case, the vector MN can be the difference between the radius vectors OM and ON. We can see why the coordinates should be subtracted in this order. Note that this theorem is true for both a plane and a three-dimensional space.
Another conclusion. If we have a segment MN with the coordinates of its ends, then the midpoint of the segment is calculated according to the specified rule: we need to consider the half-sums of the corresponding coordinates of the ends of this segment. This fact holds both for the plane and for space.
Let us prove this fact for the plane. We consider the coordinate system and the segment MN in it. We mark its midpoint, the point S. We consider the radius vectors of these points and note that the vector OS is half the diagonal of the parallelogram built on the vectors OM and ON. Then the OK vector is the sum of the OM and ON vectors, and the OS vector is half of the OK vector. Thus, we have the dependence specified in the theorem based on this vector equation.
Vectors are linearly dependent if and only if the matrix made up of their coordinates is degenerate. We can get a number of consequences derived from this fact. If we consider a plane, the collinearity of the vectors a and b is equivalent to the fact that the determinant made up of their coordinates is zero. This means that the coordinates of these vectors are proportional. For a space: the coplanarity of three vectors a, b, c is equivalent to the fact that the determinant made up of their coordinates is zero.
Now let's consider how the coordinates of the same point are related in two coordinate systems. We assume that we have a certain coordinate system and we denote it by the number 1. It is defined by the point O and two basis vectors i and j. Then we assume that there is a second coordinate system with the origin Q’ and vectors i’ and j’. First, we are going to consider the case when two bases are equally oriented, that is, the transition from one vector to another is carried out in the same direction. For example, in a counterclockwise direction. The relative position of the bases is determined by the angle between the corresponding basis vectors. Let's set all vectors from the point O, that is, from the beginning of the first coordinate system, and we are going to consider the angle between the first basis vectors, that is, between the vector i and the vector i’. Moreover, we set it in the direction that corresponds to the first coordinate system, that is, from the vector i to the vector i’.
We need to express the coordinates of the vectors i’ and j’ in the first coordinate system. The vector i’ has the coordinates (cosa, sina), which follows from the definition of trigonometric functions. The vector j’ has the following coordinates. We can find them by using the reduction formulas, since the angle between i and j’ is 90˚ plus a. We can also see this in the figure, if we consider equal right triangles with hypotenuses i’ and j’. In the drawing, equal legs are marked with corresponding strokes. We denote the coordinates of the point Q as x0 and y0. To obtain the necessary formulas, we take an arbitrary point M and consider its coordinates in different systems. In system 1, we denote the x and y coordinates, in system 2, we denote the x’, y’ coordinates. Then we consider a vector equation, that is, we express the vector OM as the specified sum and represent this vector equation in matrix form. We get the following equation: the coordinates in the first system are written to the left and right in this equation. We place the coordinates of the vectors in a column.
Now let's consider the second case, when the coordinate systems 1 and 2 are oriented oppositely, that is, the transition from the basis vectors is performed differently: in one coordinate system it is clockwise, and in the other system it i counterclockwise. We take the systems, and believe that they are oriented differently. We set all vectors from the point O. We consider the angle a. Then we find the coordinates of the vectors i’ and j’ in the first coordinate system. Using the formula of reduction, or considering appropriate equal rectangular triangles, we get the desired formula for the coordinates of the vector j’. Then we proceed in the same way. We take the point M, write its coordinates in different bases, and then present the vector equation in matrix form and get the formula that is similar to the formula obtained in the previous case.
Thus, we have two cases. They are similar, but there is a difference in the last term. However, we can write these formulas in a single form. If we introduce the coefficient e, which for the first case is equal to 1 (when the bases are equally oriented) and -1 (when the bases are oriented oppositely), we get this formula, written in matrix form. Note that a second-order matrix that consists of trigonometric functions is a transition matrix from one basis to another.
Let's write this equation in the usual form using a system of two equations, thus we get the required formula. X and y are the coordinates of the point M in the first coordinate system; x’ and y’ are the coordinates of the point M in the second coordinate system.
In conclusion, let's get acquainted with another coordinate system, where there are planes. This system is the polar coordinate system. To set it, we must select the point O (this point is the pole of this system). Then, we choose the direction, that is, the OA ray, which is the polar axis. We also need to set the orientation, and choose the orientation counterclockwise. In this case, we can specify polar coordinates for any point M.
We consider the radius vector of this point, and denote its length by the letter r. We also denote the angle between the polar axis and the vector by the letter a. Two numbers, r and a, define this point. These numbers are the polar coordinates of this point. At the same time, if we consider a point that is symmetrical with respect to the polar axis, then its first coordinate r is the same, and the second coordinate (angle a) changes to the opposite value.
Note that for the pole, that is, for the point O, the angle value is not defined, and the radius r is assumed zero.
Let us specify the relationship between the coordinates of a point in the rectangular coordinate system and in the polar coordinate system. Let's have a polar system, that is, we have a pole and a polar axis. We need to set a rectangular coordinate system. As a starting point, we choose the point O. We consider the vector i directed along the polar axis, and the vector j, which is perpendicular to this axis. Thus, we get a rectangular coordinate system with the axes Ох and Оy. We take an arbitrary point whose polar coordinates are r and a.
We use them to find rectangular coordinates. The abscissa of the point M can be expressed as r * cosa, and the ordinate of the point M is r * sina. Thus, it is easy to find rectangular coordinates using polar coordinates. We use the inverse formulas, that is, knowing the rectangular coordinates, we can specify the polar r and a. If we square the previously obtained equations and add them, we can express the number r, given that the sum of the squares of the sine and cosine is 1. To find the anglea, we need to consider the formulas: cos a=x0/r, sin a=y0/r. The angle a is selected based on these two equations.
Thus, there is a one-to-one relationship between the rectangular and polar coordinate systems.