One of enough widespread tasks, the HIGHER EDUCATION INSTITUTIONS meeting in initial courses of the higher mathematics, consists in determination of distance from any point to some plane. As a rule, the plane is set by the equation in one form or another. But there are also other methods of definition of the planes. For example, traces.
It is required to you
- - the planes given traces;
- - point coordinates.
Instruction
1. If in entry conditions do not contain the coordinate of the points which are places of crossing of the plane with axes of a system of coordinates (traces can be set in this way), define them. If traces are determined by couples of any points belonging to the XY, XZ, YZ planes work out the equations of the straight lines (in these planes) containing the corresponding pieces. Having solved the equations, find coordinates of crossings of traces with axes. Let it will be points of A(X1, Y1, Z1), B(X2, Y2, Z2), C(X3, Y3, Z3).
2. Start finding of the equation of the plane set by initial traces. Make look determinant: (X-X1) (Y-Y1) (Z-Z1)(X2-X1) (Y2-Y1) (Z2 - Z1)(X3-X1) (Y3-Y1) (Z3 - Z1) Here X1, X2, X3, Y1, Y2, Y3, Z1, Z2, Z3 - values of coordinates of points And, In, With, found on the previous step, X, Y and Z - the variables appearing in the resulting equation. Pay attention that elements of two lower lines of a matrix will contain as a result constant values.
3. Calculate determinant. Equate the received expression to zero. It will also be the plane equation. Pay attention that the determinant of a look (n11) (n12) (n13) (n21) (n22) (n23) (n31) (n32) (n33) can be calculated as: n11 * (n22*n33 - n23*n32) + n12 * (n21*n33 - n23*n31) + n13 * (n21*n32 - n22*n31). As sizes n21, n22, n23, n31, n32, n33 are constants, and the first line contains variables X, Y, Z, the resulting equation will have an appearance: AH + BY + CZ + D = 0.
4. Define distance from a point to the plane set by initial traces. Let values Xm, Ym, Zm will be coordinates of this point. Having these sizes and also coefficients And, In, With and the free member of the equation of D received on the previous step use a look formula: P = | AHm + VYm + SZm + D | / √ (And² + In² + With²) for calculation of the resulting distance.