2. Geometric representation: vectors can be represented by directed line segments. The length of the directed line segment indicates the size of the vector, and the direction pointed by the arrow indicates the direction of the vector. (If endpoint A of line segment AB is defined as the starting point and endpoint B is defined as the ending point, then the line segment has the direction and length from the starting point A to the ending point B ... This line segment with direction and length is called a directed line segment. )
3. Coordinate representation: in the plane rectangular coordinate system, two unit vectors I and J in the same direction as the X axis and the Y axis are respectively taken as the base. A is an arbitrary vector in a plane rectangular coordinate system, starting from the coordinate origin O, and the vector OP = A. According to the basic theorem of plane vectors, there are only one pair of real numbers (x, y), so that a= vector OP=xi+yj. Therefore, the pair of real numbers (x, y) is called the coordinate of vector A, and is denoted as a=(x, y). This is the coordinate expression of vector A, where (x, y) is the coordinate of point P, and vector OP is called the position vector of point P [edit this paragraph] and the size of the vector, that is, the length (or modulus) of the vector. The modulus of vector a is expressed as |a|.
note:
1, the modulus of the vector is a non-negative real number, and the sizes can be compared.
2. Because directions can't compare sizes, vectors can't compare sizes. The concepts of "greater than" and "less than" are meaningless to vectors. For example, "vector ab >；; Vector CD "is meaningless. [Edit this paragraph] Special vector unit vector
A vector with a length of 1 is called a unit vector. A vector in the same direction as the vector A with the length of 1 is called the unit vector in the direction A, and is denoted as a0, where a0=a/|a|.
Zero vector
A vector with a length of 0 is called a zero vector, and its starting point and ending point coincide, so the zero vector has no definite direction, or the direction of the zero vector is arbitrary.
Equal vector
Vectors with the same length and direction are called equal vectors. Vectors a and b are equal, so let's say a = B.
Rule: All zero vectors are equal.
When the vector is represented by a directed line segment, the starting point can be arbitrarily selected. Any two equal nonzero vectors can be represented by the same directed line segment, regardless of the starting point of the directed line segment. Directed line segments with the same direction and length all represent the same vector.
Free vector
A vector whose starting point is not fixed can move in parallel at will, and the moved vector still represents the original vector.
In the sense of free vector, equal vectors are regarded as the same vector.
Only free vectors are studied in mathematics.
Sliding vector
A vector acting along a straight line is called a sliding vector.
Fixed vector
A vector acting on a point is called a fixed vector (also called a viscous vector).
Position vector
For any point P on the coordinate plane, we call the vector OP the position vector of the point P, and write it down as follows: the vector P [edit this paragraph] whose inverse quantity is equal to the length of A and the vector whose direction is opposite is called the inverse quantity of A, and write it down as -A ... Yes-(-A) = A;
The inverse of a zero vector is still a zero vector.
Parallel vector
Non-zero vectors with the same or opposite directions are called parallel lines (or * * * lines) vectors. Vectors A and B are parallel (* * * lines), denoted as A ∨ B. 。
The length of zero vector is zero, the starting point and the ending point coincide, and the direction is uncertain. We stipulate that the zero vector is parallel to any vector.
A set of vectors parallel to the same line is a * * * line vector.
* * * quantity orientation
Three (or more) vectors parallel to the same plane are called * * * vectors.
The vector in space has and only has the following two positional relationships: (1) * * * plane; (2) not * * * face.
Only three or more vectors can talk about * * * surfaces, not to mention * * * surfaces. [Edit this paragraph] Let the operation of the vector be a=(x, y) and b=(x', y').
1, vector addition
The addition of vectors satisfies parallelogram rule and triangle rule.
AB+BC=AC .
a+b=(x+x '，y+y ').
a+0=0+a=a .
Algorithm of vector addition;
Exchange law: a+b = b+a;
Law of association: (a+b)+c=a+(b+c).
2. Vector subtraction
If a and b are mutually opposite vectors, then the reciprocal of a=-b, b=-a and a+b =0. 0 is 0.
AB-AC=CB。 That is, "* * * the starting point is the same, and the direction is reduced"
A=(x, y) b=(x', y') Then a-b=(x-x', y-y').
3. Multiply the number by the vector
The product of real number λ and vector A is a vector, which is denoted as λa, λ A ∣ = ∣ λ ∣ ∣ A ∣.
When λ > 0, λa and A are in the same direction;
When λ < 0, λa and A are in opposite directions;
When λ=0, λa=0, and the direction is arbitrary.
When a=0, there is λa=0 for any real number λ.
Note: By definition, if λa=0, then λ=0 or A = 0.
Real number λ is called the coefficient of vector A, and the geometric meaning of multiplier vector λa is to extend or compress the directed line segment representing vector A. ..
When ∣ λ ∣ > 1, the directed line segment representing vector A extends to ∣λ ∣ times in the original direction (λ > 0) or in the reverse direction (λ < 0);
When ∣ λ ∣ < 1, the directed line segment representing vector A is shortened to ∣ λ ∣ times in the original direction (λ > 0) or in the reverse direction (λ < 0).
The multiplication of numbers and vectors satisfies the following algorithm.
Law of association: (λ a) b = λ (a b) = (a λ b).
The distribution law of vector logarithm (first distribution law): (λ+μ)a=λa+μa 。
The distribution law of number pair vector (second distribution law): λ(a+b)=λa+λb 。
The elimination method of number multiplication vector: ① If the real number λ≠0 and λa=λb, then A = B. ② If a≠0 and λa=μa, then λ = μ.
4. Quantity product of vectors
Definition: Two nonzero vectors A and B are known. Let OA=a, OB=b, then the angle < a, b > is called the included angle between vector A and vector B, which is denoted as < a, b >, and is defined as 0 ≤ < A, B >≤π.
Definition: the product of two vectors (inner product, dot product) is a quantity, denoted as a B. If A and B are not * * * lines, a b = | a || b | cos < a, b >；; If a, b***, then a b =+-∣ a ∣ ∣ b ∣.
The coordinates of the product of vectors are expressed as: a b = x x'+y y'.
Vector product algorithm
A b = b a (exchange law);
(λ a) b = λ (a b) (on the associative law of number multiplication);
(a+b) c = a c+b c (distribution law);
Properties of scalar product of vectors
A a = the square of a |.
a⊥b = a b = 0 .
|a b|≤|a| |b| .
The main difference between vector product and real number operation
1, the product of vectors does not satisfy the associative law, that is, (a b) c ≠ a (b c); For example: (a b) 2 ≠ a 2 b 2.
2. The product of vectors does not satisfy the law of elimination, that is, b=c cannot be deduced from A = A = C (A ≠ 0).
3、a | b |≦| a | | b |
4. From |a|=|b|, it is impossible to deduce a=b or a =-b.
5. Cross product of vectors
Definition: the cross product (outer product, cross product) of two vectors A and B is a vector, which is denoted as a×b. If A and B are not * * * lines, the modulus of A× B is: ∣ A× B ∣ = | A || B | SIN < A, b >；; The direction of a×b is perpendicular to A and B, and A, B and a×b form a right-handed system in this order. If a and b*** line, then a×b=0.
Cross product property of vector;
∣a×b∣ is the area of a parallelogram with sides A and B.
a×a=0 .
a∨b÷a = a×b = 0 .
Cross product algorithm of vectors
a×b =-b×a；
(λa)×b =λ(a×b)= a×(λb)；
(a+b)×c=a×c+b×c。
Note: "Vector AB/ Vector CD" is meaningless without vector division.
6. Mixed product of three vectors
Definition: In a given space, three vectors A, B, C, the cross product a×b of vectors A and B, and then multiplied by vector C, the number obtained is called the mixed product of three vectors A, B, C, and recorded as (A, B, C) or (abc), that is, (ABC) = (A, B).
Hybrid products have the following characteristics:
1, the absolute value of the mixed product of three non-* * directional quantities A, B and C is equal to the volume V of a parallelepiped with edges A, B and C. When A, B and C form a right-handed system, the mixed product is positive; When A, B and C form a left-handed system, the mixed product is negative, that is, when A, B and C form a right-handed system, (ABC) = ε v (ε =1; ε=- 1) When A, B and C form a left-handed system.
2. Inference of upper property: The necessary and sufficient condition for the * * * surfaces of three vectors A, B and C is (abc)=0.
3 、( ABC)=(BCA)=(cab)=(BAC)=(CBA)=(ACB)
4 、( a×b) c=a (b×c)
Triangular inequality of vectors
1、∣∣a∣-∣b∣∣≤∣a+b∣≤∣a∣+∣b∣；
① If and only if A and B are reversed, take the equal sign on the left;
② If and only if A and B are in the same direction, the right side is an equal sign.
2、∣∣a∣-∣b∣∣≤∣a-b∣≤∣a∣+∣b∣。
① If and only if A and B are in the same direction, take the equal sign on the left;
② If and only if A and B are reversed, the right side is equal.
definite proportion
Fractional formula (vector p 1p = λ vector PP2)
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), then there is
OP =(OP 1+λOP2)( 1+λ)； (Fixed Fractional Vector Formula)
x=(x 1+λx2)/( 1+λ)，
Y=(y 1+λy2)/( 1+λ). (Fixed-point coordinate formula)
Let's call the above formula the fixed point formula of the directed line segment P 1P2.
Three-point * * line theorem
If OC=λOA +μOB, and λ+μ= 1, then the three points A, B and C are * * * lines.
Judgement formula of triangle center of gravity
In △ABC, if GA +GB +GC=O, then G is the center of gravity of △ABC [edit this paragraph]. If b≠0, then the important condition of a//b is that there is a unique real number λ, so that a = λ b.
The important condition of a/b is that xy'-x'y=0.
The zero vector 0 is parallel to any vector. [Edit this paragraph] The necessary and sufficient condition for the vertical vector a⊥b is that A ⊥ B = 0.
The necessary and sufficient condition of a⊥b is xx'+yy'=0.
The zero vector 0 is perpendicular to any vector.