1. 1 explanation

Point multiplication of vectors is also called inner product and scalar product of vectors. The dot multiplication of two vectors is the operation of summing the corresponding bits of two vectors one by one, and the result of dot multiplication is a scalar.

1.2 point multiplication formula

For vectors a(a 1, a2, …, an) and b(b 1, b2, …, bn).

a b = a 1b 1+a2 B2+…+anbn

One-dimensional vector A and vector B are required to have the same number of rows and columns.

Geometric meaning of 1.3

The geometric meaning of point multiplication can be used to describe or calculate the included angle between two vectors and the projection of B vector in the direction of A vector. There is a formula:

a b = |a||b|cosθ

So the angle between a and b vectors:

θ=arccos[(a b )/(|a||b|) ]

According to this formula, the included angle between vector A and vector B can be calculated. Therefore, we can further judge whether these two vectors are in the same direction, orthogonal (that is, vertical) and other directional relations. The specific correspondence is as follows:

AB>0 directions are basically the same, and the included angle is between 0 and 90.

A b = 0 orthogonal, perpendicular to each other.

AB<0 directions are basically opposite, and the included angle is between 90 and180.

2. Vector cross product

2. 1 explanation

The cross product of two vectors, also called cross product, outer product and cross product, is a vector rather than a scalar. The cross product direction of vector A and vector B is perpendicular to the plane where these two vectors are located, and the right-hand rule is observed. (The simple way to judge the direction of the result vector satisfying the right-handed rule is as follows: If the coordinate system satisfies the right-handed rule, when the four fingers of the right hand turn from A to B and the rotation angle does not exceed 180 degrees, the thumb points to the direction of C ...)

Its length is the area of a and b open parallelograms.