Mathematical plane vector

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').

4. Multiply the number by the vector

The product of real number λ and vector A is a vector, denoted as λ a, 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: (λa)? b=λ(a？ b)=(a？ λb)。

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.

3. Quantity product of vectors

Definition: Given two non-zero vectors A and B, let OA = A and OB = B, then the angle AOB is called the included angle between vector A and vector B, denoted as < A, B > and specified as 0 ≤

Definition: the product of two vectors (inner product, dot product) is a quantity, which is recorded as a? B. If A and B are not connected, then A? b=|a|？ |b|？ cos〈a，b〉; If a, b***, then a? b=+-∣a∣∣b∣.

Coordinate representation of vector product: a? b=x？ x'+y？ y。

Vector product algorithm

Answer? b=b？ A (commutative law);

(λa)？ b=λ(a？ B) (On the Law of Number Multiplication);

(a+b)？ c=a？ c+b？ C (distribution method);

Properties of scalar product of vectors

Answer? 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 a vector does not satisfy the law of elimination, that is, it is determined by A? b=a？ C (a≠0) and b=c cannot be deduced.

3、a？ b |≦| a |？ |b|

4. From |a|=|b|, it cannot be inferred that a=b or a =-b. 。

4. Cross product of vectors.

Definition: The cross product (outer product and 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 are * * * lines, 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×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.

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

The 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 the vector P 1P=λ? Vector PP2, λ is called the ratio of point p divided by directed line segment P 65438+P 2.

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)。 (Proportional 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.

Important conditions of vector * * * line

If b≠0, the important condition of ab is the existence of a unique real number λ, so that A = λ b. 。

The important condition of ab is xy'-x'y=0.