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高考数学平面向量公式都有哪些【一】?
作者:admin 发布于:2019-08-06 15:11 点击量:

 
 
平面向量是在二维平面内既有方向(direction)又有大小(magnitude)的量,物理学中也称作矢量,与之相对的是只有大小、没有方向的数量(标量)。平面向量用a,b,c上面加一个小箭头表示,也可以用表示向量的有向线段的起点和终点字母表示。以下是整理出来的平面向量的公式
 
数学3 (2).jpg
 
 1、向量的加法 
 
向量的加法满足平行四边形法则和三角形法则。 
AB+BC=AC。 
a+b=(x+x',y+y')。 
a+0=0+a=a。 
向量加法的运算律: 
交换律:a+b=b+a; 
结合律:(a+b)+c=a+(b+c)。 
2、向量的减法 
如果a、b是互为相反的向量,那么a=-b,b=-a,a+b=0. 0的反向量为0 
AB-AC=CB. 即“共同起点,指向被减” 
a=(x,y) b=(x',y') 则 a-b=(x-x',y-y'). 
3、数乘向量 
实数λ和向量a的乘积是一个向量,记作λa,且∣λa∣=∣λ∣?∣a∣。 
当λ>0时,λa与a同方向; 
当λ<0时,λa与a反方向; 
当λ=0时,λa=0,方向任意。 
当a=0时,对于任意实数λ,都有λa=0。 
注:按定义知,如果λa=0,那么λ=0或a=0。 
实数λ叫做向量a的系数,乘数向量λa的几何意义就是将表示向量a的有向线段伸长或压缩。 
当∣λ∣>1时,表示向量a的有向线段在原方向(λ>0)或反方向(λ<0)上伸长为原来的∣λ∣倍; 
当∣λ∣<1时,表示向量a的有向线段在原方向(λ>0)或反方向(λ<0)上缩短为原来的∣λ∣倍。 
数与向量的乘法满足下面的运算律 
结合律:(λa)?b=λ(a?b)=(a?λb)。 
向量对于数的分配律(第一分配律):(λ+μ)a=λa+μa. 
数对于向量的分配律(第二分配律):λ(a+b)=λa+λb. 
数乘向量的消去律:① 如果实数λ≠0且λa=λb,那么a=b。② 如果a≠0且λa=μa,那么λ=μ。 
4、向量的的数量积
定义:已知两个非零向量a,b.作OA=a,OB=b,则角AOB称作向量a和向量b的夹角,记作〈a,b〉并规定0≤〈a,b〉≤π
定义:两个向量的数量积(内积、点积)是一个数量,记作a?b.若a、b不共线,则a?b=|a|?|b|?cos〈a,b〉;若a、b共线,则a?b=+-∣a∣∣b∣.
向量的数量积的坐标表示:a?b=x?x'+y?y'.
向量的数量积的运算律
a?b=b?a(交换律);
(λa)?b=λ(a?b)(关于数乘法的结合律);
(a+b)?c=a?c+b?c(分配律);
向量的数量积的性质
a?a=|a|的平方.
a⊥b 〈=〉a?b=0.
|a?b|≤|a|?|b|.
向量的数量积与实数运算的主要不同点
1、向量的数量积不满足结合律,即:(a?b)?c≠a?(b?c);例如:(a?b)^2≠a^2?b^2.
2、向量的数量积不满足消去律,即:由 a?b=a?c (a≠0),推不出 b=c.
3、|a?b|≠|a|?|b|
4、由 |a|=|b| ,推不出 a=b或a=-b.
 
 
Planar vectors are quantities of both direction and magnitude in two-dimensional planes. They are also called vectors in physics, as opposed to quantities of only size and no direction. Planar vectors are represented by a small arrow on a, B and c, or by the beginning and end letters of the directed line segments representing vectors. Here are the formulas of the plane vectors sorted out.
 
Mathematics 3(2). jpg
 
1. Vector addition
 
The addition of vectors satisfies the parallelogram rule and the triangle rule.
AB + BC = AC.
A + b= (x + x', y + y').
A + 0 = 0 + a = a.
Operational Law of Vector Addition:
Exchange law: a + B = B + a;
The law of association: (a + b) + C = a + (b + c).
2. Subtraction of Vectors
If A and B are opposite vectors, then the reverse of a=-b, b=-a, a+b=0.0 is 0.
AB-AC = CB. That is "common starting point, pointing to be subtracted"
A= (x, y) b= (x', y') then a-b= (x-x', y-y').
3. Multiplier Vector
The product of real number lambda and vector a is a vector, which is denoted as lambda, and lambda=lambda_a.
When lambda > 0, lambda A and a are in the same direction.
When lambda is less than 0, lambda A and a are in the opposite direction.
When lambda = 0, lambda = 0, any direction.
When a = 0, for any real number lambda, there is lambda = 0.
Note: By definition, if lambda = 0, then lambda = 0 or a = 0.
Real number lambda is called the coefficient of vector a. The geometric meaning of multiplier vector lambda is to extend or compress the directed line segment representing vector a.
When lambda > 1, the directed line segment representing vector a extends to lambda times the original direction (lambda > 0) or the opposite direction (lambda < 0).
When lambda < 1, the directed line segment representing vector a is shortened to lambda times the original direction (lambda > 0) or the opposite direction (lambda < 0).
The multiplication of numbers and vectors satisfies the following operation laws
The law of association: (lambda) B = lambda (a b) = (a lambda b).
Distribution Law of Vector to Number (First Distribution Law): (lambda+mu) a = lambda+mua.
The law of distribution of numbers to vectors (the second law of distribution): lambda(a+b)=lambda+lambda.
Elimination law of multiplier vectors: 1. If real number lambda 0 and lambda = lambda b, then a = B. (2) If a_0 and lambda = mua, then lambda = mu.
4. Quantitative Product of Vectors
Definition: If two non-zero vectors a, B. are known as OA = A and OB = b, then the angle AOB is called the angle between vector a and vector b, which is recorded as a, B and stipulates that 0 a, B is less than pi.
Definition: The quantity product of two vectors (inner product and dot product) is a quantity, which is denoted as a B. If A and B are not collinear, then a b= | B cos a, B and if a and B are collinear, then a b=+ - a B.
The coordinates of the quantity product of vectors are expressed as a B = x x'+y y'.
Operational Law of Quantity Product of Vectors
A * b = b * a (exchange law);
(lambda) B = lambda (a b) (on the law of combination of number multiplication);
(a + b) * C = a * C + b * C (distribution law);
The Properties of the Quantity Product of Vectors
A * a= | a | squared.
A B = a B = 0.
| a * B | < | a | | B |.
The Main Differences between the Quantitative Product of Vectors and Real Number Operations
1. The number product of vectors does not satisfy the binding law, i.e. (a b) C a (b c); for example, (a b) 2 a 2 B 2.
2. The number product of vectors does not satisfy the elimination law, that is, from a B = a C (a 0), B = C can not be deduced.
3. |a.b | | a | | b|
4. From | a |= | B |, we can not deduce a = B or a= - B.
 



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