The formula of plane vector includes the arithmetic of vector addition: a+b=b+a, (a+b)+c=a+(b+c).

For two vectors A (vector a≠ vector 0) and B, when there is a real number λ, so that vector b=λ vector A (remember that vectors are directional), then vector a‖ vector B. On the contrary, when vector a‖ vector B has only one real number λ, vector b=λ vector A can be made.

Nature of quantitative products:

Given two non-zero vectors a and b, a.b = | a || b | cos θ (θ is the angle between a and b) is called the quantitative product or inner product of a and b, which is denoted as a.b. The product of zero vector and arbitrary vector is 0. The geometric meaning of product a b is the product of the length of a |a| and the projection of b in the direction of a |b|cos θ.

The product of two vectors equals the sum of the products of their corresponding coordinates. That is, if a = (x 1, y 1) and b = (x2, y2), a b = x 1 x2+y 1 y2.