Judging by calculating the mixed product of vectors: for three vectors in three-dimensional space, if their mixed product is zero, then these three vectors are * * * planes. 2/5 is judged by calculating the rank of the vector group: arrange three vectors into a matrix, and then find the rank of the matrix. If the rank of the matrix is less than or equal to 2, then these three vectors are * * * planes.

* * * Definition of quantity:

Three vectors that can be translated into a plane are called * * * directional vectors. The quantity theorem is one of the basic theorems in mathematics. It belongs to the teaching category of high school mathematics solid geometry. It is mainly used to prove two vector planes, and then prove a series of complex problems such as vertical planes.

If the two vectors a.b are not * * * lines, the necessary and sufficient condition for the vector P and the vector A. B * * plane is that there is a unique ordered real number pair (x.y), so that p=xa+yb is defined as: three vectors that can be translated to the same plane are called * * * vectors.

Inference 1:

Let O, A, B and C be four points in a non-* * plane, then there is a unique ordered real array (x, y, z) for any point P in the space, so that OP=xOA+yOB+zOC. If x+y+z= 1, PABC has four points in the * * * plane.

(1) Uniqueness: Let another set of real numbers X', Y' and Z' make OP=x'OA+y'OB+z'OC, then XOA+YOB+ZOC = X' OA+Y' OB+Z' OC ∴ (X-X').

(2) If x+y+z= 1, then PABC has four points * * * planes:

Suppose that OP=xOA+yOB+zOC and x+y+z= 1, PABC is not * * * so z= 1-x-y, then OP = XOA+YOB+(1-X-Y) OC = XOA-XOC.

Inference 2:

The necessary and sufficient condition for any point P in space to be in a plane MAB is that there exists an ordered real number pair {,x.y} such that MP=xMA+yMB or that there exists OP=OM+xMA+yMB for any given point O in space.