Geometry. Product of vectors


Let there be two vectors: \(a(x_1,y_1)\) and \( b(x_2,y_2 )\) . The area of ​​a parallelogram, "stretched" on these vectors — is the module of the skew product \(x_1 \cdot y_2-x_2 \cdot y_1\) vectors, and the area of ​​the "stretched" triangle is half that area. 
Note that the described method for finding the area is better than Heron's formula, since it does not use root extraction, which leads to loss of calculation accuracy.

Training problem

Area of ​​a triangle

Let \(C(x,y)\) be the coordinates of the point, \(A (a,b)\) - vector start coordinates, \(B(c,d)\) - vector end coordinates. First, let's find out if the point lies on the line AB! To do this, you need to calculate the skew product of the vectors AB and AC! If it is zero, then the point lies on the line! Then calculate the scalar product of the vectors AB< /code> and AC! If it is >=0 then the point belongs to the ray defined by the vector otherwise not.

Training problem

Belonging of a point to a ray