I. Introduction.
"Plane and plane perpendicular" is the core concept in high school mathematics stereo geometry, and it is also an important content to cultivate students' spatial thinking ability and logical reasoning ability. Understanding and mastering the definition, properties, judgment methods and applications of plane and plane perpendicularity are of great significance for students' subsequent mathematics learning and practical problem solving. This article will analyze the relevant knowledge points of plane and plane perpendicularity in detail to help students better grasp this content.
2. Definition of plane perpendicular to plane.
The two planes are perpendicular if and only if the dihedral angle between them is 90 degrees. A dihedral angle is a figure composed of two half-planes, the size of which is the acute angle or right angle sandwiched between the two half-planes. If the dihedral angle of the two planes is 90 degrees, the two planes are said to be perpendicular.
3. The nature of the plane perpendicular to the plane.
Dihedral corner nature: The dihedral angle of two perpendicular planes is 90 degrees, i.e. the angle between them is a right angle. This is a direct basis for judging whether the two planes are perpendicular or not.
Perpendicular nature: If a straight line is perpendicular to both intersecting planes, then the line is also perpendicular to any of the lines determined by the two planes. This property is very useful when solving geometric problems involving perpendicular lines.
Reciprocity: If two planes are perpendicular, then either line between them is perpendicular to each other. This means that all the straight lines of the two perpendicular planes are perpendicular to each other.
Fourth, the determination method of perpendicular to the plane.
Definitional Method: According to the definition of plane perpendicular to plane, if the dihedral angle of two planes is 90 degrees, then the two planes are perpendicular. This method is suitable for cases where the dihedral angle can be clearly judged to be 90 degrees.
Perpendicular method: If a line in one plane is perpendicular to the other, the two planes are perpendicular. This method is often used to solve geometric problems involving vertical relationships.
Vector method: If the normals of the two planes are perpendicular, then the two planes are perpendicular. This approach is useful when solving geometric problems involving vector operations.
5. The application of plane and plane perpendicular.
Applications in architectural designIn architectural design, designers need to take advantage of the perpendicular nature of the plane to the plane to ensure the stability and aesthetics of the building. For example, when designing the walls and floors of a building, it is necessary to ensure a vertical relationship between them to ensure that the overall structure of the building is stable and the appearance is harmonious.
Applications in engineering drawing: In engineering drawing, engineers need to take advantage of the perpendicular nature of the plane to the plane to produce accurate drawings. For example, when drawing a floor plan of a building, it is necessary to use the vertical relationship between the plane and the plane to ensure the accuracy and consistency of the drawing.
Analysis of spatial location relationships: When solving spatial geometry problems, it is often necessary to analyze the positional relationships between points, lines, and surfaces. Using the perpendicular nature of the plane to the plane can help us analyze the spatial position relationship more accurately, so as to find ideas and methods to solve the problem. For example, when judging the shape of a polyhedron, you can use the vertical relationship between planes to determine whether the faces of the polyhedron are perpendicular to each other.
Applications in physical and chemical experimentsIn physical and chemical experiments, the vertical relationship between planes is often used to describe the spatial position relationship between experimental equipment and experimental processes. For example, in optical experiments, the perpendicular relationship between the optical path and the mirror surface needs to be used to study the reflection and refraction laws of lightIn chemical experiments, it is necessary to use the vertical relationship between the various parts of the experimental setup to ensure the accuracy and safety of the experiment.
6. Summary and outlook.
Through the study of this article, students have a deeper understanding of the knowledge points of "plane and plane perpendicular". Mastering this knowledge not only helps to improve students' mathematical literacy and problem-solving skills, but also lays a solid foundation for subsequent learning and application. I hope that students will continue to consolidate and apply this knowledge point in their future studies, and explore more interesting properties and application examples related to it. At the same time, it is also expected that educators and researchers can continue to improve and expand the teaching content and methods in this field, and provide students with better educational resources and guidance. Through continuous study and practice, we believe that students will be able to master this knowledge point and apply it in real life.
New College Entrance Examination Mathematics