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高考数学平面向量公式都有哪些【二】?
作者:admin 发布于:2019-08-06 15:11 点击量:

 
 
5、向量的向量积
定义:两个向量a和b的向量积(外积、叉积)是一个向量,记作a×b.若a、b不共线,则a×b的模是:∣a×b∣=|a|?|b|?sin〈a,b〉;a×b的方向是:垂直于a和b,且a、b和a×b按这个次序构成右手系.若a、b共线,则a×b=0.
向量的向量积性质:
∣a×b∣是以a和b为边的平行四边形面积.
a×a=0.
a‖b〈=〉a×b=0.
向量的向量积运算律
a×b=-b×a;
(λa)×b=λ(a×b)=a×(λb);
(a+b)×c=a×c+b×c.
注:向量没有除法,“向量AB/向量CD”是没有意义的. 
6、向量的三角形不等式
1、∣∣a∣-∣b∣∣≤∣a+b∣≤∣a∣+∣b∣;
① 当且仅当a、b反向时,左边取等号;
② 当且仅当a、b同向时,右边取等号.
2、∣∣a∣-∣b∣∣≤∣a-b∣≤∣a∣+∣b∣.
① 当且仅当a、b同向时,左边取等号;
② 当且仅当a、b反向时,右边取等号.
7、定比分点
定比分点公式(向量P1P=λ?向量PP2)
设P1、P2是直线上的两点,P是l上不同于P1、P2的任意一点.则存在一个实数 λ,使 向量P1P=λ?向量PP2,λ叫做点P分有向线段P1P2所成的比.
若P1(x1,y1),P2(x2,y2),P(x,y),则有
OP=(OP1+λOP2)(1+λ);(定比分点向量公式)
x=(x1+λx2)/(1+λ),
y=(y1+λy2)/(1+λ).(定比分点坐标公式)
我们把上面的式子叫做有向线段P1P2的定比分点公式
 
8、三点共线定理 
若OC=λOA+μOB,且λ+μ=1 ,则A、B、C三点共线 
三角形重心判断式 
在△ABC中,若GA+GB+GC=O,则G为△ABC的重心 
[编辑本段]向量共线的重要条件 
若b≠0,则a//b的重要条件是存在唯一实数λ,使a=λb。 
a//b的重要条件是 xy'-x'y=0。 
零向量0平行于任何向量。 
[编辑本段]向量垂直的充要条件 
a⊥b的充要条件是a?b=0。 
a⊥b的充要条件是xx'+yy'=0。 
零向量0垂直于任何向量.
 
 
 
5. Vector Product of Vectors
Definition: The vector product (outer product and cross product) of two vectors a and B is a vector, which is recorded as a *b. If A and B are not collinear, the modulus of a *b is a * b = | a | | B | sin a, B and the direction of a B is perpendicular to a and b, and a, B and a B constitute a right-handed system in this order. If A and B are collinear, a B = 0.
Vector product properties of vectors:
_a *b is a parallelogram area with edges a and B.
A * a = 0.
A B = a B = 0.
Vector Product Operational Law of Vectors
A * b = - B * a;
(lambda)*b=lambda(a*b)=a*b;
(a+b)*c=a*c+b*c.
Note: Vectors do not divide and "vector AB/vector CD" is meaningless.
6. Vector Triangular Inequality
1. _a-b<a+b<a+b;;
(1) If and only if a and B are reversed, the left side is equal sign;
(2) If and only if a and B are in the same direction, take the equal sign on the right.
2. _a-b= a-b= a-b= a_a+b.
(1) If and only if a and B are in the same direction, the left side takes the equal sign;
(2) If and only if a and B are reversed, take the equal sign on the right.
7. Fixed Score Point
Fixed Ratio Point Formula (Vector P1P = Lambda * Vector PP2)
Let P1 and P2 be two points on a straight line and P be any point on L different from P1 and P2. Then there exists a real number lambda, so that vector P1P = lambda vector PP2, lambda is called the ratio of point P to directed line segment P1P2.
If P1 (x1, y1), P2 (x2, y2), P (x, y), then there are
OP= (OP1+lambda OP2) (1+lambda); (Fixed Ratio Scorepoint Vector Formula)
X=(x1+lambda x 2)/(1+lambda),
Y=(y1+lambda y2)/(1+lambda). (Fixed Ratio Point Coordinate Formula)
Let's call the above formula the fixed fraction formula of the directed line segment P1P2.
 
8. Three-point collinearity theorem
If OC = lambda OA + mu OB and lambda + Mu = 1, then A, B and C are collinear
The Judgment Formula of the Center of Gravity of a Triangle
In Delta ABC, if GA+GB+GC=O, then G is the center of gravity of Delta ABC.
Important Conditions for Vector Colinearity
If b_0, the important condition of a//b is the existence of a unique real number lambda, so that a = lambda.
The important condition of a//b is x y'-x'y=0.
Zero vector 0 is parallel to any vector.
Necessary and Sufficient Conditions for Vector Verticality
The necessary and sufficient condition for a B is a B = 0.
The necessary and sufficient condition for a_b is xx'+yy'=0.
Zero vector 0 is perpendicular to any vector.
 



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