Modulus length of vector
The modulus length of a vector indicates the length or size of the vector, that is, the length of a straight line segment between the starting point and the ending point of the vector. In vector quantity product, the modulus length of vector is used to calculate the numerical part, that is, the result of multiplication operation.
Cosine value of included angle
The cosine of the included angle refers to the cosine corresponding to the included angle between two vectors. In cross product, the cosine of the included angle is used to measure the correlation or similarity between two vectors. When the included angle is a right angle, the cosine value is 0; When the included angle is acute, the cosine value is positive; When the included angle is obtuse, the cosine value is negative.
The result of vector product
The result of vector quantity product is a real number, which represents some relationship or property between two vectors. Specifically, the result of vector quantity product is equal to the product of the modulus length of two vectors and the cosine value of the included angle. This result can be used to calculate the projection of vectors, judge the directional relationship between two vectors, and solve plane geometry problems.
Geometric meaning
Direction relation: By calculating the cosine of the included angle, the direction relation between two vectors can be judged. When the included angle is 0 degrees, the result of vector quantity product is the product of the module length of two vectors, indicating that the two vectors have the same direction; When the included angle is 180 degrees, the result of vector quantity product is the product of two negative vector module lengths, indicating that the two vectors are in opposite directions.
Projection relation: the product of vectors can be used to calculate the projection length of one vector on another vector, that is, the component of one vector in the direction of another vector. This projection length can help us understand the projection properties of vectors and related geometric problems.
Orthogonality: When two vectors are perpendicular, the cosine of the included angle is 0, and the result of vector quantity product is 0. So we can use the product of vectors to judge whether two vectors are vertical or orthogonal.
Expand knowledge:
Vector product has a wide range of applications in geometry, such as calculating the angle between two straight lines on the plane, calculating the area of a triangle, and solving the projection of vectors in space. In addition, the product of vectors is closely related to the inner product and outer product of vectors, which is an important concept in linear algebra and vector analysis.