There is a set of ways to set the same plane in space - with use of coordinates of points in various coordinate systems, the indication of the general, initial or parametrical equations of the plane. It is possible for this purpose to use vectors, the equations of straight lines and curves and also different combinations of all listed options. Below is given a little bit most often used methods.
Instruction
1. Set the plane the indication of coordinates of three incoincident points belonging to the set of points making the plane. An indispensable condition which at the same time has to be executed - the specified points should not lie on one straight line. For example, you can safely claim that there is a plane which unambiguously decides by points on coordinates of A(8,13,2) B (1,4,7) C (-3,5,12).
2. Other method - a plane task by means of the equation is more widely used. In a general view it looks so: Ax + By + Cz + D = 0. Coefficients of A, B, C, D can be calculated on coordinates of points, having made matrixes for each of them and having calculated determinants. For coefficient of A place three coordinates of three points in which all abscissae are replaced with unit in every line of a matrix. For coefficients it is necessary to replace with B and the C units, respectively, ordinate and z-coordinate, and for a matrix of coefficient D it is not necessary to change anything. Having calculated a determinant of each matrix, substitute them in the general equation of the plane, having changed at the same time the sign of coefficient of D. For example, for the example given in the previous step the formula has to have such appearance: - 50*x + 15*y - 43*z + 291 = 0.
3. For a plane task instead of three points it is possible to use one point and a straight line as two points in space unambiguously define the only straight line. To use this method, a point specify it by three-dimensional coordinates, and a straight line - the equation. The equation is written in a general view so: Ax + By + C = 0. For the example used above the plane can be set coordinates of a point of C (-3,5,12) and the equation of a straight line 2x - y + z - 5 = 0 - it is received from coordinates of points of A and B.
4. Instead of the equation of direct coordinate of a point it is possible to add with coordinates of a normal vector - this couple of data will set the unique plane too. For the plane A point with coordinates (8,13,2) and a vector ō (-50.15,-43 can make such couple of examples of the previous steps).
5. It is possible to set the plane and couple of crossed or parallel straight lines. In this case give their standard or initial equations. For the same example it is possible to set the plane in couple of the equations of straight lines on which couples of points of A,B and A, C lie: 2x - y + z - 5 = 0 and - 18x + 11y - 11z - 19 = 0.