is the cross product of a two-dimensional vector a scalar or a vector?

Today, I learned a bit about computational geometry, which is about determining whether a point is within a triangle (on a two-dimensional plane). One of the algorithms is "the same direction method", mainly with the cross product to determine whether the two points on the same side of a line segment, (1) is shown. About "The same way" again not to do specific introduction, interested students can Baidu, or focus on my post updated. I'll continue to learn, summarize and post on the blog in the "Computational Geometry" series.

Figure 1

Well, let's go to the next level, we only know that in a two-dimensional plane, the cross of two vectors is the exact value of the result (cross product). For example vector A (x1,y1) and Vector B (x2,y2) fork Multiply: A (x1,y1) XB (x2,y2) = | a| | B|sina, where a is the angle between vector A and vector B, | A| and | B| is a modulo of vector A and B, and Sina is a real number related to angle a. You can see that the final result of vector A and vector b is an exact real number. The cross product of two vectors that we know is a normal vector perpendicular to its plane, because we get a value, not a vector. This is God horse situation?

For a triangular ABC, the angle of the CA and CB is a, then the area of the triangle s = |ca| | Cb|sina), which is S = CAXCB, can be regarded as the area of the triangle. By the right-hand coordinate system, we know that the direction is positive, then back to our side is negative, because for a plane, there must be a positive and negative. If the direction n of the planar normal vector of the triangle is defined, then n is perpendicular to the vector CA,CB and is consistent with the normal vector of the triangle's front. Therefore, the cross product of a two-dimensional vector cannot be simply regarded as a scalar, which is essentially a vector.

The following two-dimensional vector as a z-axis value of a constant 0 three-dimensional vector, such as Vector OA (x1,y1,0), OB (x2,y2,0), which is essentially a planar xy vector, but the introduction of z-axis, so that it has three-dimensional meaning of space. The value of Oaxob can be calculated based on the cross product formula (1-1) of the three-dimensional vector.

(1-1)

Its matrix is represented as follows:

Then Oaxob = (0,0,x1y2-x2y1)(#1), it is obvious (0,0,x1y2-x2y1) is also a vector, and is perpendicular to the vector OA and the plane formed by the normal vector, and the direction of the positive and negative depends on their input (x1,y1) and ( The value of the X2,y2).

is the cross product of a two-dimensional vector a scalar or a vector?