Foundation of Image Processing-Inner Product and outer product

1. The inner product of the vector is the quantity product of the vector. The angle between the two non-zero vectors is counted as <A, B>, and <A, B >ε [0, π]. Definition: The number product (Inner Product and dot product) of two vectors is a quantity, which is recorded as a · B. If A and B are not collocated, A · B = | A | · | B | · cos <A, B>. If a and B are collocated, then, a · B = +-then a then splits into B then. 2. Vector's outer product is the vector's Vector Product Definition: the Vector Product (outer product, Cross Product) of two vectors A and B is a vector, recorded as a × B. If A and B are not in the same line, the modulo of a x B is as follows: running a x B then = | A | · | B | · sin <A, B> 〉; the direction of a x B is vertical to A and B, and A, B and A x B constitute the right hand system in this order. If A and B are both linear, a x B = 0.

 

If the inner product of two vectors is zero, the two vectors are vertical;

The outer product is 0, indicating that the two vectors are parallel.

 

 

A. B In the vector indicates that the coordinate values of the two vectors are multiplied and then added. a scalar number is obtained.

The application of the cosine ry where the value is equal to the length product of two vectors multiplied by the angle: The angle between two vectors can be obtained;

The geometric meaning of the inner product of one vector A and one unit vector e is the projection vector of A in the E direction.

The length of a vector after the square of its inner product is

What we get from a × B of Vector's outer product is a vector, a determinant. Taking 3D vectors as an example, equal to | I j k | A1 A2 A3 | B1 B2 B3 | the length value is equal to the sine of the length product of the two vectors multiplied by the angle. The direction is determined by the right hand spiral, physical applications are often applied to electromagnetic force ry: the outer product of two vectors is equal to the parallelogram area with two vectors as the adjacent edge, and the direction is the normal direction of the plane where the two vectors are located;

Point multiplication and cross Multiplication

15:07:15 | category: vector | font size subscription

Mathematical formula of dot multiplication and cross Multiplication

 

Let's take a look at the mathematical definitions of aspect multiplication and cross multiplication:

Point multiplication is also called the Inner Product and number product of a vector. As the name suggests, the result is a number.

Dot Product: (x1, Y1, Z1). (X2, Y2, Z2) = x1x2 + y1y2 + z1z2

The dot product can be used to calculate the angle between two vectors. The formula is as follows:

Cos (V ^ W) = V. W/| v | w |

Cross multiplication is also called the outer product and vector product of a vector. As the name suggests, the result is a vector, and the vector is C.

| Vector c | = | vector A × vector B | = | A | B | sin <A, B>

Cross multiplication: (x1, Y1, Z1) x (X2, Y2, Z2) = (y1z2-z1y2, z1x2-x1z2, x1y2-y1x2)

The vertical vector of the two vectors can be calculated by the cross-Multiplication. The new vector after the cross-multiplication is the vector perpendicular to the first two vectors.

The direction of vector C is perpendicular to the plane where A and B are located, and the direction must be determined by the "right hand rule" (the four fingers in the right hand first represent the direction of vector, then, the finger moves toward the palm of the hand to the direction of vector B, and the thumb points to the direction of vector C ).

Therefore

The outer product of a vector does not conform to the multiplication exchange rate because

Vector a × vector B =-vector B × vector

In physics, it is known that the moment of force and force arm is the outer product of the vector, that is, the cross multiplication.

Represent the vector with coordinates (3D vector ),

If vector A = (A1, B1, C1), vector B = (A2, B2, C2 ),

Then

Vector A · vector B = a1a2 + b1b2 + c1c2

Vector a × vector B =

| I j k |

| A1 B1 C1 |

| A2 B2 C2 |

= (B1c2-b2c1, c1a2-a1c2, a1b2-a2b1)

(I, j, and K are the unit vectors of the three axes perpendicular to each other in the space ).

The geometric meaning of the Cross multiplication is: for example, if the vector X Cross multiplication vector Y is the right hand four pointing to the same direction as X to Y, the direction of the thumb is the direction of the Cross multiplication result, the size is the area enclosed by the parallelogram on the adjacent sides of the vector X and Y.

The geometric meaning of point multiplication is that the projection of an edge to another edge is multiplied by the length of another edge.

 

Application of crossover:

We know that the coordinates of the three vertices of the Triangle ABC are a (-) B (6,-2) C () to calculate the Triangle Area.

Vector AB = 8i-2j, vector AC = 3I + 4.7

Triangle ABC area = | vector AB cross multiplication vector AC |/2

= | (8i-2j) Cross multiplication (3i + 4.7) |/2

= | 48 k + 6 k |/2

= | 54k |/2

= 27, (I, j, and K are the three units of the spatial Cartesian coordinate system)

 

Http://blog.csdn.net/carina197834/article/details/8185902

Foundation of Image Processing-Inner Product and outer product