Vectors locating points in planes and space

The discussion for planes is similar to the discussion for lines. It requires two points to determine a (straight) line, but the points cannot coincide. It requires three points to determine a plane, but the points cannot lie on the same line (not collinear). Later the following statement will be examined: it requires four points to determine space, but no three of them can lie in the same plane (not coplanar).

Let A and B be any two distinct points. A point P lies on the line AB if an only if there is a real number λ satisfhying the vector equation   AP = λAB.   Let A, B and C be any distinct points, such that they do not all lie on the same line. Then point P lies in the plane ABC determined by the three points if and only if there exist real numbers λ and σ satisfying the vector equation   AP = λAB + σAC. This is an addition of vectors, and therefore involves a involves a parallelogram. Let B^ and C^ be points on lines AB and AC such that AB^ = λAB and AC^ = σAC. Then AB^PC^ is an addition parallelobram.