Construction of a Plane From an Angular Vector

The procedure of constructions of a plane from an angular vector in a rectangular coordinate system has been described. The method of creating of an angular vector from two rectilinear vectors has been considered. The property of equality of angular vectors has been described.


Introduction
With the advent of angular vectors in the vector theory (Kharchenko, 2017), there was a need for their use not only for modeling angular physical quantities, but also for geometric constructions. Angular vectors can be used as oriented planes in the coordinate system. These vectors allow us to perform operations similar to rectilinear vectors, such as: constructing parallel and perpendicular planes; creating a plane perpendicular to the two original planes; construction of a rectilinear vector at the intersection of two planes, etc. One of the first and important steps for these operations is to solve the problem of constructing a plane from an angular vector.
Before deriving a method for constructions of a plane from an angular vector and for a better understanding of the properties of vectors, we will consider: 1.
method of creating an angular vector from two rectilinear vectors, 2.
property of the equality between angular vectors.

Creating an Angular Vector
The equation linking rectilinear and angular vectors is the cross product equation . This equation shows not only the relations between vectors, but also the method of creating an angular vector from two rectilinear vectors.
The equation shows that an angular vector is created from a sequence of two rectilinear (perpendicular to each other) vectors. The sequence of rectilinear vectors determines the direction of an angular vector.
In geometric form, when creating an angular vector for the equation , it is convenient to imagine that rectilinear vector begins to be created b at the end of the rectilinear vector a . And the position of these rectilinear (perpendicular) vectors, sets the direction of the angular vector с  , Fig. 1.
The direction sign of an angular vector, determines the angular direction in the selected coordinate system, just like the directions of the coordinate axes in the coordinate system determine the sign of rectilinear vector.
The value of an angular vector is equal to the area of the rectangle formed from rectilinear vectors. For all variants of rectangular coordinate systems (and there are 96 types = 8 octants of dividing the space * 6 variants of the location of the axis names in each octant * 2 variants of the angular direction for each version of the coordinate system), only two versions of the equations for finding the projections of angular vectors can be derived.
For a positive angular direction between the axes Oz Ox Oy   (the direction can be selected both clockwise and counterclockwise, and depends on the selected coordinate system): For a positive angular direction between the axles Oz Oy Ox   (the direction can be selected both clockwise and counterclockwise, and depends on the selected coordinate system): This option of coordinating the direction of angular vectors (and their projections) in various coordinate systems is slightly different from that proposed earlier in "Angular Vectors in the Theory of Vectors". This description is the best, since it completely solves the problem of choosing the angular direction in the coordinate system and the equations for finding projections of an angular vector.
For the cross product equation , you do not need to use the right-hand rule (in mathematics) and the right-hand screw rule (in theoretical mechanics), which greatly simplifies the solution of problems and increases the choice of the coordinate system.

Property of the Equality Between Angular Vectors
Let's remember that two rectilinear vectors b a , are equal when they: have equal modules, are equally oriented relative to the coordinate axes (they have the same projection values on the corresponding coordinate axes), directions of vectors coincide (signs of all projections of the vectors on coordinate axes coincide).
When writing of rectilinear vectors in algebraic form, vectors are equal But, despite the similar properties of the equality of rectilinear and angular vectors, there are differences between them. They appear during the geometric construction of vectors, Fig. 3.
Equal rectilinear vectors are always parallel and codirectional in space (coordinate system).
For equal angular vectors, only the planes in which they are located are parallel. But the location of an angular vector in the plane (the location of the rectilinear vectors from which an angular vector is built) can be any. And the shape of the resulting the area figure (rectangles or squares) can be any. Moreover, the areas of equal angular vectors are equal to the modules of angular vectors.
From the equality of angular vectors it follows that the same angular vector can be represented from various combinations of rectilinear vectors. Therefore, angular vectors are often displayed conditionally (without a geometric figure).

Figure 3. Equality of rectilinear and angular vectors
It also follows from the equality of angular vectors that the projection of an angular vector onto any other plane (including the coordinate one) does not depend on the shape and location of an angular vector on the plane in which it is located.

The Method of Construction of a Plane From an Angular Vector
It is difficult to construction of an angular vector and its projections onto the coordinate planes in a coordinate system. After all, we only have the area values of the figures on these planes (projections of an angular vector), and we do not know their shapes and locations in the coordinate planes. And to construction of them, we need the original two rectilinear vectors. Their location, size, and direction determine the location, direction, and shape of an angular vector.
Based on the property of the equality of angular vectors, we can represent an angular vector from any sequence of two rectilinear vectors.
Let's imagine that the angular vector ) , , ( . This condition determines the dependence of the projection values of two vectors b a , , which allow them to be placed so that they begin and end on the coordinate axes, and the signs allow us to solve the radical expression (3). , we find the equation for finding the projections of the angular vector from the cross product equation (1).
We find the projections of rectilinear vectors from these equations.  Vol. 12, No. 2;2020 In order to find a pair of vectors, we used the cross product equation . In fact, we do not know the sequence of rectilinear vectors in the cross product that suits us (for the same angular vector c  ). If we chose the cross product c a b    , we would get pairs of vectors:

Let's choose from the obtained four vectors a suitable pair of vectors by an additional condition
Next, we find the coordinates of points A, B, C. Each pair of rectilinear vectors obtained can be used to construct an angular vector. But we, by the condition of the problem, need to construction of a plane with points located on the coordinate axes. To do this, we can re-find rectilinear vectors using the cross product

Conclusion
The described simple procedure shows that construction of a plane from an angular vector allows us to find several solutions at once (to create several parallel planes) that expand the capabilities of engineers in geometric constructions.
The considered method of forming an angular vector from two rectilinear vectors shows a geometric relationship between these vectors. This explanation of the logical dependence of rectilinear and angular vectors is the missing link in proving the existence of the cross product equation for vectors с b a    . The considered property of the equality of angular vectors allows us to expand the methods of solving problems and when solving them, visually represent the behavior of the angular vector and its projections.

Refutation
In the paper "Angular Vectors in the Theory of Vectors", the projections of an angular vector onto the coordinate planes were connected by a sequence of indices