Let a=(x, y) and b=(x', y'). The addition of 1. 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 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, "* * * is the same starting point and the number of points to be subtracted" a=(x, y) b=(x', y'). When λ > 0, λa and A are in the same direction; 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 ∣ 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 operation rules: (λa)b=λ(ab)=(aλb). 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 \. 3. Definition of vector product: Two non-zero vectors A and B are known. Let OA = A and OB = B, then the angle AOB is called the included angle between the vector A and the vector B, which is denoted as < a, b > and defined as 0 ≤ < A, and B ≤π: the quantitative product (inner product, dot product) of the two vectors is a quantity, which is denoted as ab. If a and b are not * * * lines, AB = | A || B | COS < A, B >；; If lines A and b***, then ab=+-∣a∣∣b∣. The coordinate of the product of vectors is ab=xx'+yy'. The algorithm of vector product ab=ba (exchange rule); (λa)b=λ(ab) (on the associative law of number multiplication); (a+b)c=ac+bc (distribution law); Properties of the product of the square of vector aa = a |. A⊥b÷ab = 0 .|ab|≤|a||b|. The main difference between vector product and real number operation is 1, and vector product does not satisfy the associative law, namely: (ab) c ≠ a (BC); For example: (ab) 2 ≠ A 2b 2. 2. The product of vectors does not satisfy the law of elimination, that is, b=c cannot be deduced from ab=ac (a≠0). 3.| AB |≦| A | | | B | 4。 From |a|=|b|, a=b or A =-B. 4 cannot be deduced. Definition of cross product of vectors: the cross product (outer product and cross product) of two vectors A and B is a vector, 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. The cross product property of vectors: ∣a×b∣ is the area of a parallelogram with side lengths A and B, and A× A = 0. A× B ‖ = A× B = 0. The cross product algorithm of vectors A× B =-B× A; (λa)×b =λ(a×b)= a×(λb)； (a+b) × c = a× c+b× C Note: It is meaningless for vector AB/ vector CD not to be divided. Triangular inequality of vector 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 constant fraction point formula (vector P 1P=λ vector PP2) makes P 1 and P2 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 = (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. 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 ab=0. The necessary and sufficient condition of a⊥b is xx'+yy'=0. The zero vector 0 is perpendicular to any vector. Copy:/question/110272005. HTML Responder: 312776268 | Level 5 | 2011-2-2715: 366. Let OA=a and OB=b, then the angle AOB is called the included angle between the vector A and the vector B, which is denoted as < a, b > and defined as 0 ≤ < A and B ≤π: the product (inner product, dot product) of the two vectors is a quantity, which is denoted as ab. If a and b are not * * * lines, AB = | A || B | COS < A, B >；; If lines A and b***, then AB =+-∣ A ∣∣ B. (positive in the same direction and negative in the opposite direction). The coordinate representation of the product of vectors: ab=xx'+yy'. The algorithm of vector product ab=ba (exchange rule); (λa)b=λ(ab) (on the associative law of number multiplication); (a+b)c=ac+bc (distribution law); Properties of the product of the square of vector aa = a |. A⊥b÷ab = 0 .|ab|≤|a||b|. The main difference between vector product and real number operation is 1, and vector product does not satisfy the associative law, that is, (AB) C ≠ A (BC); For example: (ab) 2 ≠ A 2b 2. 2. The product of vectors does not satisfy the law of elimination, that is, b=c cannot be deduced from ab=ac (a≠0). 3.| AB |≤| |ab|≤|a||b| 4。 A=b or A =-B cannot be deduced from |a|=|b|. The cross product of vectors | AXB | = | A || B | SIN < a, b>.