In BC 146, the Romans conquered Greece. In 47 BC, Caesar set fire to the Egyptian fleet anchored in Alexandria. The fire spread to the city and ruthlessly destroyed the books collected by the library for two and a half centuries. The spread of Christianity, which was highly praised by Roman rulers, quickly drowned the rich scientific imagination with strong religious fanaticism, which made Greek mathematics suffer even greater disasters. The closure of the academy and the prohibition of learning and studying mathematics made European mathematics enter a long dark period. /kloc-In the 5th century, with the collapse of Byzantine Empire, refugees fled to Italy with their wealth including ancient Greek culture. From the middle of15th century to the end of16th century, this period is called the Renaissance in Europe. During this period, Europe began to see great ideological liberation, great production development and great social progress, and science and culture, including mathematics, began to recover and prosper. By the17th century, the capitalist relations of production from within the feudal society were on the rise, which promoted the rapid development of social productive forces. Ocean navigation, mining, machinery manufacturing and the external expansion of capital have raised many problems for natural science, such as the movement of celestial bodies, the swing of clocks and watches, the trajectory of artillery shells and the shape of lenses, which are beyond the scope of Euclidean geometry. Analytic geometry founded by Fermat and Descartes solved the above problems. Analytic geometry is the product of the combination of algebra and geometry. By introducing coordinate system into geometry, geometric problems are transformed into algebraic problems, and variables are introduced into mathematics, which makes it possible for people to quantitatively analyze the law of motion change with the help of mathematics. Maurice Klein, a famous American mathematical historian, pointed out: "As long as algebra and geometry go their separate ways, their progress will be slow and their application will be narrow. But when these two kinds of science are combined into a partner, they draw fresh vitality from each other, and since then, they have gone to perfection at a rapid speed. " /kloc-In the first half of the 7th century, mathematicians accumulated a lot of knowledge and methods of calculus, and the appearance of analytic geometry laid the foundation for the establishment of calculus. As Engels said: "The turning point in mathematics is Descartes' variable; With variables, movement enters mathematics; With variables, dialectics enters mathematics; With variables, differentiation and integration are immediately needed. "
In analytic geometry, we can prove some theorems by constructing vectors or simplify the process of proving some theorems.
Using the scalar product, cross product and mixed product operations in spatial analytic geometry, the mixed product of a vector and three non-scalar factorizations is operated, and then the coordinate representation of the mixed product is applied to the spatial right-hand Cartesian coordinate system. After substituting the coordinates of four vectors, we can prove the important theorem of solving linear equations in linear algebra-Cramer's rule.
Cauchy-Schwartz inequality is an important inequality in mathematical analysis, which can be proved by the definition of scalar product and the coordinate expression of scalar product in spatial rectangular coordinate system. We can also prove two important equations in mathematical analysis-Lagrange identity and Jacobian identity by using the calculation formula of binary product.
After constructing a vector in a triangle, we can prove the cosine theorem in trigonometry by using the definition of product of quantity and the algorithm, and we can also prove another theorem in trigonometry-sine theorem by using the definition of cross product module.