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.

Note

The scalar product is a scalar characteristic of the lengths of vectors and angle between them.