EquationOfAPlane.md (1621B)
1 # Equation of a Plane 2 3 Khan 4 5 **Definition:** The equation of a plane is the equation that defines all points on the plane as a combination of n variables where n is the number of dimensions we are in. This is the definition of plane when in 3d space and a hyperplane in higher dimensions. 6 7 ### Plane Formula 8 9 The general formula for a 3d plane is as follows: 10 11 ax + by + cz = d 12 13 where a,b,c, and d are coeficcients and x,y, and z are variables. 14 15 ### Hyperplane formula 16 17 A hyperplane in n dimensional space is defined as follows: 18 19 a_1 x_1 + a_2 x_2 + ... + a_n x_n = d 20 21 Where, again, all a sub values are coeficcients along with d and all x values are variables. 22 23 ### Calculation 24 25 When we have a normal vector and a point this is very simple. 26 27 Steps: 28 29 1. Plug in normal vector for equation of the plane: 30 - ax + by + cz = d where a,b,c are the x,y, and z axis components of the vector 31 2. Plug in representative point as x,y, and z and then solve for d 32 33 This can be extrapolated into higher dimensions for hyper-planes assuming we still have a representative point and the normal vector. 34 35 ### No Normal Vector 36 37 To find the normal vector of a plane when we just have three points or two vectors and one point we can find the normal vector as follows: 38 39 1. If we have three points then find two vectors on the plane by taking the difference between a reference point and the two other points on the plane. If these vectors are colinear (dependent) then we don't have enough information to get the formula for the plane. 40 2. Take the cross product of both vectors to find the normal vector 41 3. Complete calculation steps above