The concept of plane vector, a compulsory knowledge point of mathematics in college entrance examination;
(1) vector: a quantity with both magnitude and direction. Vectors can't compare sizes, but the modules of vectors can compare sizes.
(2) Zero vector: a vector with a length of 0, which is recorded as 0, with arbitrary direction, and 0 is parallel to any vector.
(3) Unit vector: a vector with the modulus of 1 unit length.
(4) Parallel vectors: non-zero vectors with the same or opposite directions.
(5) Equal vectors: vectors with equal length and the same direction.
Plane Vector Product Analysis of Required Knowledge Points in Mathematics of College Entrance Examination
1, product of plane vector quantity: two non-zero vectors a and b are known, so |a||b|cos? (? Is the angle between a and b) is called the product or inner product of a and b, and is recorded as a? B the product of zero vector and arbitrary vector is 0. Quantity product a? The geometric meaning of b is: the length of a |a| and the projection of b in the direction of a |b|cos? The product of.
The product of two vectors equals the sum of the products of their corresponding coordinates. That is, if a = (x 1, y 1) and b = (x2, y2), then a? b=x 1？ x2+y 1？ y2
2. The product of the plane vector has the following properties:
1、a？ a=|a|2？ 0
2、a？ b=b？ a
3、k(a？ b)=(ka)b=a(kb
4、a？ (b+c)=a？ b+a？ c
5、a？ B = 0<=> A? b
6、a = kb & lt= & gta//b
7、e 1？ e2=|e 1||e2|cos？
Analysis of plane vector addition of required knowledge points in college entrance examination mathematics
Given the directional quantities AB and BC, and then the vector AC, then the vector AC is called the sum of AB and BC, and marked AB+BC, that is AB+BC=AC.
Note: The addition of vectors satisfies all the laws of addition, such as commutative law and associative law.
Plane Vector Subtraction Analysis of Required Knowledge Points of Mathematics in College Entrance Examination
1, AB-AC=CB, this calculation rule is called the triangle rule of vector subtraction, abbreviated as: * * * * starting point and finger subtraction.
-(-a)= a； a+(-a)=(-a)+a = 0； a-b=a+(-b).
Overview of plane vector formulas
1, set the score point
Formula of cut-off point (vector P 1P= 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. And then there is a real number? , so that vector P 1P= vector PP2,? It 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=(OP 1+？ OP2)( 1+？ ); (Fixed Fractional Vector Formula)
x=(x 1+？ x2)/( 1+？ ),
y=(y 1+？ y2)/( 1+？ )。 (Fixed-point coordinate formula)
Let's call the above formula the fixed point formula of the directed line segment P 1P2.
2. Three-point * * * line theorem
If OC=? OA +？ OB, then what? +? = 1, then the three-point * * line of A, B and C.
Judgement formula of triangle center of gravity
In △ABC, if GA +GB +GC=O, then G is the center of gravity of △ABC.
[Edit this paragraph] Important conditions of vector * * * line
If b? The important condition of 0, A/B is the existence of a unique real number? Make an =? B.
The important condition of a/b is that xy'-x'y=0.
The zero vector 0 is parallel to any vector.
[Edit this paragraph] Necessary and Sufficient Conditions for Vector Verticality
Answer? Is b a necessary and sufficient condition? b=0 .
Answer? The necessary and sufficient condition of b is xx'+yy'=0.
The zero vector 0 is perpendicular to any vector.
Let a=(x, y) and b=(x', y').
3. 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).
4. 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。 What is that? * * * The same starting point, the direction is lowered?
A=(x, y) b=(x', y') Then a-b=(x-x', y-y').
5. Multiply the number by the vector
Real number? The product of vector a is a vector, which is recorded as? ∣? a∣=∣？ ∣? ∣a∣。
What time? & gt at 0 o'clock? A is in the same direction as a;
What time? & lt at 0 o'clock? A is opposite to a;
What time? =0,? A=0, any direction.
When a=0, for any real number? , both? a=0 .
Note: By definition, if? So A=0? =0 or a=0.
Real number? A coefficient is called a vector, a multiplier vector? The geometric meaning of a is to extend or compress the directed line segment representing vector a.
What time? ∣>； 1, the directed line segment of vector a is in the original direction (? & gt0) or in the opposite direction (? & lt0) The upper elongation is the original ∣? Times;
What time? ∣<； 1, the directed line segment of vector a is in the original direction (? & gt0) or in the opposite direction (? & lt0) shortened to the original ∣? Time magazine.
The multiplication of numbers and vectors satisfies the following algorithm.
Law of association: (? a)？ b=？ (a？ b)=(ab).
Distribution law of vector logarithm (first distribution law): (? +? )a=？ a+？ a.
Distribution law of number pair vector (second distribution law):? (a+b)=？ a+？ b.
Elimination method of number multiplication vector: ① If the real number is 0 and? a=？ B, then a = B. 2 If A? 0 and? a=？ A so? =? .
6. Quantity product of vectors
Definition: Two nonzero vectors A and B are known. Let OA=a and OB=b, then the angle < a, b > is called the included angle between vector A and vector B, which is denoted as < a, b > and designated as 0? "A, B"
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 * * * lines, 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？ Yes.
Vector product algorithm
Answer? b=b？ A (commutative law);
(? a)？ b=？ (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 |.
Answer? b÷a = a？ b=0 .
|a？ b|？ | One |? |b| .
7. The main difference between vector product and real number operation.
The product of (1) vector does not satisfy the associative law, that is, (a? b)？ c？ Answer? (b？ c)； For example: (a? b)^2？ a^2？ b^2。
(2) The product of vectors does not satisfy the law of elimination, that is, it is determined by A? b=a？ c (a？ 0), b=c cannot be deduced.
(3)|a？ b|？ | One |? |b|
(4) From |a|=|b|, it is impossible to deduce a=b or a =-b.
8. 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. If A and B are not * * * lines, then A? The module of B is: ∣a? b∣=|a|？ |b|？ sin〈a，b〉; Answer? The direction of b is perpendicular to a and b, a, b and a? B constitutes a right-handed system in this order. If a, b***, then a? b=0 .
Cross product property of (1) vector;
∣a？ B∣ is the area of a parallelogram with side lengths A and B.
Answer? a=0 .
a‖b÷a = a？ b=0 .
(2) The algorithm of cross product of vectors.
Answer? b=-b？ a；
(? a)？ b=？ (a？ b)=a？ (? b)；
(a+b)？ c=a？ c+b？ c.
Note: There is no division for vectors. Vector AB/ vector CD? Meaningless.
(3) Triangle inequality of vectors
∣∣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.
∣∣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;