How to find the angle between the vectors

A vector is a segment with the given direction. The angle between the vectors has a physical meaning, for example when finding the length of the projection of the vector on the axis.

Instruction

1. The angle between two nonzero vectors is determined by calculating the scalar product. By definition, the scalar product equals the product of the lengths of vectors on a cosine of the angle between them. On the other hand, the scalar product for two vectors a with coordinates (x1; y1) and b with coordinates (x2; y2) is calculated according to the formula: ab = x1x2 + y1y2. Of these two methods of finding the scalar product it is easy to find the angle between the vectors.
2. Find the length or the modules of the vectors. For our vectors a and b: |a| = (x12 + y12)^1/2, |b| = (x22 + y22)^1/2.
3. Find the scalar product of vectors, multiplying their coordinates by pairs: ab = x1x2 + y1y2. From the definition of the scalar product ab = |a|*|b|*cos α, where α is the angle between the vectors. Then we get that x1x2 + y1y2 = |a|*|b|*cos α. Then cos α = (x1x2 + y1y2)/(|a|*|b|) = (x1x2 + y1y2)/((x12 + y12)(x22 + y22))^1/2.
4. Find the angle α using the tables Bradis.
5. In the case of three-dimensional space adds a third coordinate. For vectors a (x1; y1; z1) and b (x2; y2; z2), the formula for the cosine of the angle is shown in Fig.