Basic knowledge of mathematical vector

Basic knowledge of mathematical vectors

1. Vector related definitions

2. The linear operation of vectors

3. Vector product and quantity product

   The difference between vector product and quantity product

name	Scalar product / Inner product / The product of quantity / Dot product	Vector product / Exoproduct / Vector product / Cross product
Arithmetic expression （a,b and c In bold , It's a vector ）	a·b=|a||b|·cosθ	a×b=c, among |c|=|a||b|·sinθ,c The direction of follows the right-hand rule
Geometric meaning	vector a In vector b Projection and vector in direction b Product of modules of	c It's vertical a、b On the plane , And in
The difference of operation results	Scalar （ Often used in Physics ）/ Number （ Often used in Mathematics ）	vector （ Often used in Physics ）/ vector （ Often used in Mathematics ）

3.1 Vector product

   The vector product can be defined as ：

   Die length ：（ ad locum θ Represents the angle between two vectors （ Under the premise of common starting point ）（0°≤θ≤180°）, It lies on the plane defined by these two vectors .）

   Direction ： a Vector and b The vector product of a vector c The direction of is perpendicular to the plane of these two vectors , Follow the right-hand rule （ Right hand rule ： Use the four fingers of the right hand to represent the vector first a The direction of , Then the finger swings towards the palm to the vector b The direction of , The direction pointed by the thumb is the vector c The direction of . ）

   The formula ： set up a=(x₁,y₁,z₁),b=(x₁₂,y₂,z₂),i,j,k Namely X,Y,Z Unit vector in the axial direction , be axb It's the same as （ With the help of third-order determinant operation , That's how it's defined ）. From the result, we can see that the basic unit of the vector is still preserved i、j、k, So the result is also a vector , Existing size , There's a direction ：axb=(y₁z₂-z₁y₂,z₁x₂-x₁z₂,x₁y₂-y₁x₂)

   Two dimensional vector cross multiplication formula a（x1,y1）,b（x2,y2）, be a×b＝（x1y2－x2y1）, What does not need to be proved is the defined operation .
   Three dimensional cross multiplication is a determinant operation , It is also the definition of cross product , You think of the third dimension as 0 Just insert it .

   Geometric meaning ：

  With vectors a Sum vector b Form a parallelogram , Then the module length of the outer product of these two vectors is equal to the area of the parallelogram .

3.1 The product of quantity

    Vector point multiplication （ Inner product ）：

    Point multiplication （Dot Product） The result of is dot product , Also known as quantitative product or scalar product （Scalar Product）, The result is a number , The direction is erased . A vector has two properties ： Size and direction , The result of dot multiplication is a scalar . It is equivalent to dimension reduction .

    Defined as ： The operation of multiplying and adding the values at the corresponding positions of two vectors , The result is the dot product .

    From this result, we can see that , I know there is no direction attribute , Just running between numbers , The final result is also a number .

    Geometrically , The dot product is the product of the length of two vectors and the cosine of their angle . The result of dot multiplication represents a vector A In vector B Projection and vector in direction B Product of modules

    The meaning of dot multiplication ：

• The result of the common accumulation of two vectors in one vector direction , But this result only retains the size attribute , The attribute of direction is erased ;
• At the same time, it reflects the similarity of the two vectors in the direction , The bigger the results, the more similar ;
• Based on the result, we can judge whether the two vectors are in the same direction , Whether it is orthogonal or vertical , The specific correspondence is ：

    Greater than 0. Then the direction is basically the same , What's the angle 0° To 90° Between

   0 Then orthogonal , perpendicular

    Less than 0 The direction is basically the opposite , What's the angle 90° To 180° Between

    The difference between vertical and orthogonal ：

    Normal vector and plane

    The normal vector ： The vector perpendicular to the plane is called the normal vector of the plane (normal vector)

    Suppose a fixed point on the plane is , Any other point on the plane is , The normal vector of the plane is ：

    vector On a flat surface , And with the normal vector vertical , therefore ：

   R3 The general form of the median plane is ：

    summary ： Normal vector and a fixed point on the plane , You can define this plane