1, vector addition

The addition of vectors satisfies parallelogram rule and triangle rule. Vector addition

AB+BC=AC .a+b=(x+x', Y+Y'). A+0 = 0+A = A. Algorithm of vector addition: exchange rule: 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, "* * * has the same starting point and points to the subtraction of vectors.

Subtract "a=(x, y)b=(x', y') and 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, λa ∣ = ∣ λ A ∣. When λ > 0, λ A and A are in the same direction; When λ < 0, λa and A are in opposite directions; vector multiplication

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. The real number λ is called the coefficient of vector A, and the geometric meaning of the 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 opposite direction (λ < 0); When ∣∣∣ < 1, the directed line segment representing vector A is shortened to ∣∣∣ times in the original direction (λ > 0) or in the opposite direction of xx (λ < 0). The multiplication of numbers and vectors satisfies the following operation rules: (λ a) b = λ (a b) = (a λ b). Distribution law of vector logarithm (first distribution law): (λ+μ) a = λ a+μ a. Distribution law of number pair vector (second distribution law): λ (a+b) = λ a+λ b. Elimination law of number multiplication vector: ① If real number λ≠0 and λa=λb, then a = ② If a \.

4. Quantity product of vectors

Definition: Two nonzero vectors A and B are known. Let OA=a and OB=b, then the angle AOB is called the included angle between vector A and vector B, which is denoted as < a, B >；, and it is defined that 0 ≤ < A, and B ≤π: the product of two vectors (inner product, dot product) is a quantity, which is denoted as A B. If A and B are not * * straight lines, AB = ||| |. If a, b*** lines, then a b =+-∣ a ∣ ∣ b ∣. The coordinate of the product of the vector is expressed as a b = x x'+y y'. The algorithm of a vector's product A B = B A (exchange rule); (λ a) b = λ (a b) (on the associative law of number multiplication); (a+b) c = a c+b c (distribution law); The property of the product of vector a = the square of | a |. A⊥b = a b = 0 .|a b|≤|a| |b|. (The formula is proved as follows: | AB | = | A ||| B ||||| Cosα | Because 0≤|cosα|≤ 1, | AB |≤| A |. 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|, we can't 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, denoted as a×b (this is not a multiplication symbol, but a representation, which is different from "∧"). 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. The cross product property of vectors: ∣a×b∣ is the area of a parallelogram with side lengths A and B, a×a=0 .a vertical b < = > a× b = | a || b |. Cross product algorithm of vector A× B =-B× A; (λa)×b =λ(a×b)= a×(λb)； A× (b+c) = a× b+a× C Note: It is meaningless for vector AB/ vector CD to be divided.

6. Mixed product of three vectors

Definition: Give three vectors A, B and C in a given space, the cross product a×b of vectors A and B, and then make a quantity product (A× B) C with vector C, the mixed product of vectors.

The obtained number is called the mixed product of three vectors A, B and C, and it is recorded as (A, B, C) or (abc), that is, the mixed product of (ABC) = (A× B) C has the following properties: 1, three non-* * dimensions A, B and C. When A, B and C form a left-handed system, Inference of upper property: the necessary and sufficient conditions for the * * * plane of three vectors A, B and C are (ABC) = 0 3, (ABC) = (BCA) =-(BAC) =-(CBA) =1

7. Binary product of three vectors

Because the calculation of the two-way cross product is complicated, the simplified formula and proof process are given directly below: simplified formula and proof of the two-way cross product.

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.

The formula of fixed point (vector P 1p = λ vector PP2) assumes that P 1 and P2 are two points on a straight line, and p is any point on L different from p 1 and P2. Then there is an arbitrary real number λ and λ is not equal to-1, 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 = (op1+λ op2)/(1+λ). (Fixed point vector formula) x = (x 1+λ x2)/( 1+λ), y = (y 1+λ y2)/( 1+λ). (Fixed-point coordinate formula) We call the above formula the fixed-point formula of directed line segment P 1P2. Three-point * * line theorem If OC=λOA +μOB and λ μ =1,then the triangle barycenter judgment formula of A, B and C is △ABC, if GA+GB+GC.

Conditions of vector * * * line

If b≠0, the important condition of a//b is the existence of a unique real number λ, which makes a = λ B. If a=(x 1, y 1) and b=(x2, y2), then x1y2 = x2y/kloc-. The zero vector 0 is parallel to any vector.

Necessary and Sufficient Conditions for Verticality of Vector

The necessary and sufficient condition for a⊥b is that a b = 0, that is, x 1x2+y 1y2=0. The zero vector 0 is perpendicular to any vector. Decomposition Theorem of Plane Vector Decomposition Theorem of Plane Vector: If e 1 and e2 are two nonparallel vectors on the same plane, then for any vector on this plane, there is only one pair of real numbers λ 1, λ2 makes a = λ 65438+e 1+λ 2e2. Let's take the nonparallel vector E65438.