Dot and cross product

The inner product of a vector （ Point multiplication ）

The result is a scalar （ The set of real Numbers ）

Definition

In a nutshell , Vectorial Inner product （ Point multiplication / The product of quantity ）. Dot multiply two vectors , It is the sum operation after multiplying the corresponding bits of these two vectors one by one , As shown below , For vector a Sum vector b：

a and b The dot product formula of is ：

This requires a one-dimensional vector a Sum vector b The number of rows and columns is the same . Be careful ： The result of dot multiplication is a scalar ( Quantity, not vector )

Definition ： Two vectors a And b The inner product of is  a·b = |a||b|cos∠(a, b), Specially ,0·a =a·0 = 0; if a,b Is a nonzero vector , be a And b**** orthogonal If and only if a·b = 0.

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The outer product of a vector （ Cross riding ）

The result is a vector

Definition

In a nutshell , The outer product of two vectors , Also called cross multiplication 、 Cross product vector product , The result of the operation is vector Instead of a scalar . And the outer product of the two vectors is perpendicular to the coordinate plane composed of the two vectors .

Definition ： vector a And b Outer product of a×b It's a vector , Its length is equal to |a×b| = |a||b|sin∠(a,b), Its direction is orthogonal to a And b. also ,(a,b,a×b) Form a right-handed system .
Specially ,0×a = a×0 = 0. Besides , For any vector a,a×a=0.

For vector a Sum vector b：

a and b The outer product formula of is ：

among ：

according to i、j、k Inter relationship , Yes ：

The geometric meaning of vector outer product

In 3D Geometry , vector a Sum vector b The result of the outer product of is a vector , A more common name is normal vector , The vector is perpendicular to a and b A plane of vectors .

stay 3D In iconography , The concept of outer product is very useful , You can use the outer product of two vectors , Generate a third perpendicular to a,b The normal vector of , To build X、Y、Z Coordinate system .