Computation Ry-dot product/Cross Product

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Author: Beiyu http://www.cnblogs.com/beiyuoi/.

Reprinted please keep this text

Preface

I personally summarized the concept and application of dot product cross product qwq.

Update record

20160521-Point product/Cross Product define basic application properties of geometric meaning operations

20160609-Brief introduction-toys

Point product

Dot product, also name Point product, number product, High School Mathematics Textbook compulsory four vector mentioned

• Definition

\ [A \ cdot B = \ left | A \ right | \ times \ left | B \ right | cos <A, B> \]

• Geometric meaning \ (A \) modulus of the positive projection multiplication \ (B \) on \ (B \)
• Coordinate operation

\ [A (X _ {1}, Y _ {1}), B (X _ {2}, Y _ {2 }) => A \ cdot B = X _ {1} X _ {2} + Y _ {1} y _ {2} \]

• Length and angle of the application calculation \ (A \) and \ (B \)
• The Nature satisfies the exchange law of various laws, the combination Law, the distribution law or something.
• Note that the result of \ (A \ cdot B \) is a value rather than a vector.

 

Cross Product

Cross product, also known as Vector Product

• Definition

\ (P_1 \ times P_2 = \ begin {vmatrix} X_1 & X_2 \ Y_1 & Y_2 \ end {vmatrix} = x_1y_2-x_2y_1 \)

\ (P_1 \ times P_2 = | P_1 | \ cdot | P_2 | \ sin \ Alpha \)

• Quadrilateral area enclosed by geometric meaning vector \ (P_1 P_2 \)
• See the definition of coordinate operations.
• Calculate the perimeter of a polygon and determine the intersection of a straight line and the relationship between them on the left or right of the straight line.
• Properties meet various rules \ (P_1 \ times P_2 =-P_2 \ times P_1 \)
• Determine the linear relationship

If \ (P_1 \ times P_2 <0 \), \ (P_1 \) is in \ (P_2 \) counterclockwise direction;

If \ (P_1 \ times P_2> 0 \), then \ (P_1 \) is in the clockwise direction of \ (P_2;

If \ (P_1 \ times P_2 = 0 \), \ (P_1 P_2 \) is collocated.

 

Brief Introduction

• Poj 2318 toys (Computational ry + cross product + binary)

Give n straight lines that do not want to be handed in, divide the plane into n + 1 areas, give m points, and ask about the number of points in each area.

SOL: for the region where each vertex belongs, it is obvious that the region is monotonic. Use the cross product to determine whether the vertex is on the left or right of the straight line. The judgment method is to judge the positional relationship between the two straight lines formed by the above and the two endpoints of the straight line. If it is on the left,Upper Boundary and pointAndBottom Boundary and pointAnd vice versa.

Computation Ry-dot product/Cross Product