Find the normal vector of the plane passing through points a (0,0,0), B (1,4,0), C (0,2,0)

There are two methods
1、 Let the equation of plane ABC be ax + by + CZ + D = 0,
Three equations can be obtained by substituting three-point coordinates
D=0 ；
A+4B+D=0 ；
2B+D=0 ,
Taking a = b = D = 0, C = 1, the equation of plane is Z = 0,
So the normal vector can be (0,0,1)
2、 Because AB = (1,4,0), AC = (0,2,0),
Try to find the vector n = (x, y, z), from ab * n = 0, ac * n = 0,
We get x + 4Y = 0, 2Y = 0,
Taking x = y = 0, z = 1, the normal vector is n = (0,0,1)

If the normal vector of plane a is m = (1,0-1), and the normal vector of plane B is n = (0, - 1,1), then the two faces formed by plane a and plane B are

2 π / 3 or π / 3

How to determine the direction of normal vector of space plane?

The normal vector of a space plane can be obtained by coordinate method or geometric method. The coordinate method is to select a suitable point as the origin of the space geometric figure, obtain the coordinates of the points on the plane according to the size, and then obtain the vector form of the line. The normal vector can be obtained by multiplying the normal vector point by the line vector in the plane as 0

In the space coordinate system, three coordinates are known. When the three coordinates form a surface, how to find the normal vector of the surface
Such as the title

Let three points be a, B and C, then the vector AB and the vector AC can be solved. (either AB, AC or BC is OK) let the normal vector be a = (x, y, z), then the vector a is multiplied by the vector AB as 0, and the vector a is multiplied by the vector AC as 0, then the vector a can be solved

Find the normal vector of the plane passing through points a (0,0,0), B (1,4,0), C (0,2,0)