The orthogonal decomposition and coordinate representation of plane vectors is an important part of high school mathematics vectors, which is of great significance for understanding the properties and applications of vectors. In this article, we will analyze in detail the definition, properties, and operation rules of orthogonal decomposition and coordinate representation of plane vectors and their applications in mathematics, so as to help readers better grasp this key knowledge point.
1. Orthogonal decomposition of plane vectors.
Definition: Orthogonal decomposition of a plane vector refers to the process of decomposing a plane vector into two component vectors perpendicular to each other. These two component vectors correspond to the projections of the original vectors on the x-axis and y-axis, respectively.
Properties of orthogonal decomposition:
1) Uniqueness: For a given plane vector and coordinate system, its orthogonal decomposition is unique.
2) Perpendicular to each other: Two component vectors are perpendicular to each other, i.e. their quantity product is zero.
The operation rule of orthogonal decomposition: let the projections of the plane vector a on the x-axis and y-axis a1 and a2 be a1 and a2 respectively, then a = a1 + a2, and a1 is parallel to the x-axis and a2 is parallel to the y-axis.
2. Coordinate representation of plane vectors.
Definition: In a planar Cartesian coordinate system, any plane vector can be represented by a pair of real numbers, which are called the coordinates of the vector. In general, we take the length of the projection on the x-axis as the abscissa and the length of the projection on the y-axis as the ordinate.
Nature of Coordinate Representation:
1) Order: The coordinates of the vector are ordered pairs of real numbers, and the order cannot be reversed.
2) Equality: Two vectors are equal if and only if their coordinates are equal.
The operation rule of coordinate representation: let the coordinates of the plane vector a be (x1, y1), the coordinates of vector b be (x2, y2), then the coordinates of a + b are (x1 + x2, y1 + y2), and the coordinates of a are (x1, y1).
3. Application of orthogonal decomposition and coordinate representation in mathematics.
Addition and subtraction of vectors: Through orthogonal decomposition and coordinate representation, we can convert the addition and subtraction of vectors into the addition and subtraction of coordinates, thus simplifying the calculation process.
Quantity product and vector product of vectors: Using orthogonal decomposition and coordinate representation, we can easily calculate the quantity product and vector product of two vectors, and then judge the angle and vertical relationship between the two vectors.
Application problems of plane vectors: Orthogonal decomposition and coordinate representation are important tools to solve the application problems of plane vectors. For example, in physics, we can use orthogonal decomposition and coordinate representation to analyze problems such as the synthesis and decomposition of forces, the motion of objects, and so on.
4. Summary and expansion.
The orthogonal decomposition and coordinate representation of plane vectors are important contents in high school mathematics, and it is of great significance to understand the nature and application of vectors by mastering their definitions, properties, operation rules and their applications in mathematics. In the learning process, attention should be paid to understanding the geometric and physical significance of orthogonal decomposition and coordinate representation, and the knowledge learned should be consolidated through a lot of practice and practice. At the same time, we can try to apply the concepts and methods of orthogonal decomposition and coordinate representation to problem solving in other fields to deepen our understanding and application ability. High School Mathematics Knowledge Sharing