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The quantity that specifies the direction and size is called a vector. Vector, also known as vector, was originally applied to physics. Many physical quantities such as force, velocity, displacement, electric field intensity and magnetic induction intensity are vectors. Around 350 BC, Aristotle, a famous ancient Greek scholar, knew that force can be expressed as a vector, and the resultant force of two forces can be obtained through the famous parallelogram rule. The word "vector" comes from directed line segments in mechanics and analytic geometry. Use directed lines for the first time.
The origin of vectors
Vector, also known as vector, was originally applied to physics. Many physical quantities such as force, velocity, displacement, electric field intensity and magnetic induction intensity are vectors. Around 350 BC, Aristotle, a famous ancient Greek scholar, knew that force can be expressed as a vector, and the resultant force of two forces can be obtained through the famous parallelogram rule. The word "vector" comes from directed line segments in mechanics and analytic geometry. Newton, a great British scientist, was the first to use directed line segments to represent vectors.
The vector discussed in the textbook is a quantity with geometric properties. In addition to the zero vector, you can always draw an arrow to indicate the direction. However, there are more vectors in advanced mathematics. For example, if all polynomials with real coefficients are regarded as a polynomial space, the polynomials here can be regarded as a vector. In this case, it is impossible to find the starting point and the ending point, or even draw an arrow to indicate the direction. The vectors in this space are much wider than those in geometry. It can be any mathematical object or physical object. This can guide the application of linear algebra method to a wide range of natural science fields. Therefore, the concept of vector space has become the most basic concept in mathematics and the central content of linear algebra, and its theories and methods have been widely used in various fields of natural science. Vector and its linear operation also provide a concrete model for the abstract concept of "vector space".
Judging from the history of mathematical development, the vector structure of space has not been recognized by mathematicians for a long time in history. It was not until the end of 19 and the beginning of the 20th century that people linked the nature of space with vector operation, making vector a mathematical system with excellent universality of operation.
Vector can enter mathematics and develop, first of all, we should start with the geometric representation of complex numbers. At the end of18th century, wiesel, a Norwegian surveyor, first expressed the complex number A+Bi with points on the coordinate plane, and defined the vector operation with geometric complex operation. Points on the coordinate plane are represented by vectors, and the geometric representation of vectors is used to study geometric problems and trigonometric problems. People gradually accepted complex numbers and learned to use them to represent and study vectors on the plane.
But the use of complex numbers is limited, because they can only be used to represent planes. If there are forces that are not in the same plane acting on the same object, we need to find the so-called three-dimensional "complex number" and the corresponding operation system. /kloc-In the middle of 0/9th century, British mathematician Hamilton invented quaternion (including quantity part and vector part) to represent vectors in space. His work laid the foundation for vector algebra and vector analysis. Subsequently, the electromagnetic theory was established.
The initiation of three-dimensional vector analysis and the formal division of quaternion were independently completed by Gubbs and Hiveside in Britain in the 1980s of 19. They put forward that vector is only the vector part of quaternion, but it is not independent of any quaternion. They introduced two kinds of multiplication, namely product and cross product of quantities, and extended vector algebra to vector calculus with variable vectors. Since then, vector method has been introduced into analytical and analytic geometry.
Application of vector
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In mathematics, we usually use points to indicate position and rays to indicate direction. In a plane, all rays from any point can be used to represent all directions in the plane.
A vector is usually represented by a directed line segment, the length of which indicates the size of the vector, and the direction indicated by the arrow indicates the direction of the vector.
Vector can also be represented by letters A (1), b, c, etc. Or use letters to indicate the starting point and ending point of the vector directed segment.
The size of the vector, that is, the length (or modulus) of the vector is recorded as |a| A vector with a length of 0 is called a zero vector, and a vector with a length equal to 1 unit length is called a unit vector.
Parallel vector and equal vector
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Non-zero vectors with the same or opposite directions are called parallel vectors. The vectors A, B and C are parallel, and it is marked as A ∨ B ∨ C.0. The vector length is zero, the starting point and the ending point coincide, and the direction is uncertain. We specify that 0 is parallel to any vector.
Vectors with the same length and direction are called equal vectors. Vectors A and B are equal, so let's say A = B, and zero vector and zero vector are equal. Any two equal nonzero vectors can be represented by the same directed line segment, regardless of the starting point of the directed line segment.
vector operation
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1, vector addition:
AB+BC=AC
Let a=(x, y) b=(x', y')
Then a+b=(x+x', y+y')
The addition of vectors satisfies parallelogram rule and triangle rule.
Properties of vector addition:
Exchange method:
a+b=b+a
Association rule:
(a+b)+c=a+(b+c)
a+0=0+a=a
2. Vector subtraction
AB-AC=CB
a-b=(x-x '，y-y ')
If a//b
Then a=eb
Then xy'-x' y = 0.
If a is perpendicular to b
Then ab=0
Then xx`+yy`=0.
3. Vector multiplication
Let a=(x, y) b=(x', y')
A b (dot product) = = x x'+y y' = | a || b | * cos included angle
Let P 1 and P2 be two points on a straight line, and p is any point on L different from P 1 and P2. Then there is a real number λ, so that the vector p 1p=λ vector pp2, and λ is called the ratio of point p to directed line segment P 1P2.
If p 1 (x 1, y 1), p2 (x2, y2), p (x, y)
x=(x 1+λx2)/( 1+λ)
Then {
y=(y 1+λy2)/( 1+λ)
Let's call the above formula the fixed point formula of the directed line segment P 1P2.
4. Multiply the number by the vector
The product of the real number ∮ and the vector A is a vector, which is marked as ∮a and ∣∮∣ = ∣∮∣ * ∣ A ∣, and when ∮ > 0, it is in the same direction as A; When ∮ < 0, it is opposite to A.
The real number ∮ is called the coefficient of vector A, and the geometric meaning of the multiplier vector enlarges or narrows the direction or the opposite direction of vector A. ..
Vector, also known as vector, was originally applied to physics. Many physical quantities such as force, velocity, displacement, electric field intensity and magnetic induction intensity are vectors. Around 350 BC, Aristotle, a famous ancient Greek scholar, knew that force can be expressed as a vector, and the resultant force of two forces can be obtained through the famous parallelogram rule. The word "vector" comes from directed line segments in mechanics and analytic geometry. Newton, a great British scientist, was the first to use directed line segments to represent vectors.
The vector discussed in the textbook is a quantity with geometric properties. In addition to the zero vector, you can always draw an arrow to indicate the direction. However, there are more vectors in advanced mathematics. For example, if all polynomials with real coefficients are regarded as a polynomial space, the polynomials here can be regarded as a vector. In this case, it is impossible to find the starting point and the ending point, or even draw an arrow to indicate the direction. The vectors in this space are much wider than those in geometry. It can be any mathematical object or physical object. This can guide the application of linear algebra method to a wide range of natural science fields. Therefore, the concept of vector space has become the most basic concept in mathematics and the central content of linear algebra, and its theories and methods have been widely used in various fields of natural science. Vector and its linear operation also provide a concrete model for the abstract concept of "vector space".
Judging from the history of mathematical development, the vector structure of space has not been recognized by mathematicians for a long time in history. It was not until the end of 19 and the beginning of the 20th century that people linked the nature of space with vector operation, making vector a mathematical system with excellent universality of operation.
Vector can enter mathematics and develop, first of all, we should start with the geometric representation of complex numbers. At the end of18th century, wiesel, a Norwegian surveyor, first expressed the complex number A+Bi with points on the coordinate plane, and defined the vector operation with geometric complex operation. Points on the coordinate plane are represented by vectors, and the geometric representation of vectors is used to study geometric problems and trigonometric problems. People gradually accepted complex numbers and learned to use them to represent and study vectors on the plane.
But the use of complex numbers is limited, because they can only be used to represent planes. If there are forces that are not in the same plane acting on the same object, we need to find the so-called three-dimensional "complex number" and the corresponding operation system. /kloc-In the middle of 0/9th century, British mathematician Hamilton invented quaternion (including quantity part and vector part) to represent vectors in space. His work laid the foundation for vector algebra and vector analysis. Subsequently, the electromagnetic theory was established.
The initiation of three-dimensional vector analysis and the formal division of quaternion were independently completed by Gubbs and Hiveside in Britain in the 1980s of 19. They put forward that vector is only the vector part of quaternion, but it is not independent of any quaternion. They introduced two kinds of multiplication, namely product and cross product of quantities, and extended vector algebra to vector calculus with variable vectors. Since then, vector method has been introduced into analytical and analytic geometry.