Directed line segment: a directed line segment is called a directed line segment, and a directed line segment with A as the starting point and B as the ending point is marked as or AB;
Modulus of vector: The length of the directed line segment AB is called the module of vector, and it is recorded as | AB |.
Zero vector: A vector with a length equal to 0 is called a zero vector and recorded as or 0. (Note the bold format. There is a difference between real number "0" and vector "0". When writing, add an arrow to the real number "0" to avoid confusion);
Equal vector: vectors with the same length and direction are called equal vectors;
Parallel vector (* * * line vector): two non-zero vectors with the same or opposite directions are called parallel vectors or * * * line vectors, and the zero vector is parallel to any vector, that is, 0//a;
Unit vector: A vector with a modulus equal to 1 unit length is called a unit vector, usually represented by E, and unit vectors parallel to the coordinate axis are customarily represented by I and J, respectively.
Inverse quantity: A vector with the same length and opposite direction is called the inverse quantity of A, —(—a)=a, and the inverse quantity of a zero vector is still a zero vector.
2. Plane vector operation
Algebraic operations of addition and subtraction;
(1) if a=(x 1, y 1), b=(x2, y2), a b=(x 1+x2, y 1+y2).
Geometric representation of vector addition and subtraction: parallelogram rule and triangle rule.
Vector addition has the following laws:+=+(commutative law); +(+c)=(+)+c (law of association);
Product of real number and vector: The product of real number and vector is a vector.
( 1)| |=| | | |;
(2) when a >; 0, which is the same as direction A; When a<0, it is opposite to the direction of a; When a=0, a=0.
Necessary and sufficient conditions for two vector lines;
The necessary and sufficient condition for the straight line between (1) vector b and non-zero vector * * * is that there is only one real number, so b=.
(2) If = () and b= (), then ‖ b.
3. The basic theorem of plane vector
If e 1 and e2 are two nonlinear vectors on the same plane, there is only one pair of real numbers for any vector on this plane, so = e 1+ e2.
4. Relevant inference of plane vector.
A point O in the triangle ABC, OA ob = ob oc = oc OA, is the vertical center of the triangle.
If O is the outer center of triangle ABC and point M satisfies OA+OB+OC=OM, then M is the vertical center of triangle ABC.
If O and the triangle ABC*** surface satisfy OA+OB+OC=0, then O is the center of gravity of the triangle ABC.
Three-point * * * line: three-point A, B and C*** lines are derived from OA=μOB+aOC(μ+a= 1).
Compulsory Four Mathematics Chapter Two Theorems of Knowledge Points 21 and 22
1, * * * line vector theorem;
The * * * lines of two vectors (parallel) are equivalent to the fact that the two vectors satisfy the number multiplication relation (the vector multiplied by the real number is not a zero vector), and the number multiplication coefficient is unique. In the form of coordinates, two vector lines mean that the inner product of two vector coordinates is equal to the outer product. This theorem can be used to prove that vectors are parallel, or to make use of the condition of polarity. The extension of this theorem is a three-point * * * line! A three-point * * * line can be transformed into an equation of two vectors: 1. Arbitrarily find two vector * * * lines composed of two groups of points from three points, and satisfy the multiplication relation; 2. Take the same point as the starting point and three points as the ending point to construct three vectors, one of which can be expressed linearly by the other two, and the sum of the coefficients is 1.
2, the basic theorem of plane vector:
The vectors of two non-* * lines on the plane can linearly represent any vector with unique coefficients. These two vectors that are not * * * lines form a set of bases, and these two vectors are called basis vectors. This theorem has two functions: 1 Can unify the form of vectors in the topic; 2. The uniqueness of the coefficient can be used to find the coefficient of the vector (fixed algorithm mode).
Second, three forms
There are three forms of plane vector: letter form, geometric form and coordinate form. Attention should be paid to the letter form with arrows, and more attention should be paid to geometric drawing to solve problems, especially when special triangles and quadrilaterals can be obtained. The coordinates of vectors and points should not be confused. The coordinates of a vector are the coordinates of its endpoint minus the coordinates of its starting point. In special cases, if the starting point is at the origin, the coordinates of the vector are the end point.
Choosing the appropriate vector form is a key to solving problems. Drawing mainly in geometric form, followed by coordinate form, and finally considering the deformation operation in letter form.
Three or four operations
Addition, subtraction, multiplication and product. The first three operations are linear operations, and the result is a vector (0 times any vector to get zero vector, and zero vector times any real number to get zero vector); The product of quantities is not a linear operation, and the result is a real number (zero vector multiplied by any vector is zero). Linear operation conforms to all the laws of real number operation, and the product of quantities does not conform to the laws of elimination and combination.
There are also three forms of vector operation: letter form, geometric form and coordinate form.
The letter forms of addition and subtraction pay attention to the end-to-end connection, and the starting point coincides. The letter formula of quantity product is very important and should be used skillfully and flexibly.
The geometric meaning of addition and subtraction is the law of parallelogram and triangle, the geometric meaning of number multiplication is the expansion and contraction of lines in length and direction, and the geometric meaning of quantity product is the modulus of one vector multiplied by the projection number of another vector in the first vector direction. The included angle of vectors is indicated by angle brackets, which is the angle formed when the starting point or ending point of two vectors coincide, and the angle formed by the end-to-end connection is the complementary angle of the included angle of vectors. There are two solutions to the projective number: 1. Multiply the modulus of the vector by the cosine of the included angle; 2. The product of two vectors divided by the modulus of another vector.
The coordinate forms of addition and subtraction are the addition and subtraction of abscissa and ordinate, the coordinate form of number multiplication is the real number multiplied by abscissa and ordinate, and the coordinate form of quantity product is the product of abscissa and ordinate.
Four or five applications
Find the relationship between the length, angle, vertical, parallel, sum and difference products of vectors and the sum and difference products of modules. The first three applications are the operational properties of scalar product, which proves the operational properties of parallel line multiplication. The zero vector cannot be said to be the same or opposite to any vector direction, and it is stipulated that the zero vector is parallel and perpendicular to any vector. A vector multiplied by itself and then squared is the length; The product of the number of two vectors divided by the modulus is the cosine of the included angle; If two vectors satisfy the multiplication relation, they must be * * * lines (parallel). A vector divided by its own module gets the unit vector in the same direction as itself, and the symbol is the unit vector in the opposite direction.
The range and maximum knowledge point of mathematical function
The range of 1. function depends on the defined range and the corresponding law. No matter what method is used to find the function range, we should first consider defining the range. The common methods to find the range of functions are as follows:
(1) direct method: also known as observation method, for functions with simple structure, the range of the function can be directly observed by applying the properties of inequality to the analytical expression of the function.
(2) Substitution method: A given complex variable function is transformed into another simple function re-evaluation domain by algebraic or trigonometric substitution. If the resolution function contains a radical, algebraic substitution is used when the radical is linear and trigonometric substitution is used when the radical is quadratic.
(3) Inverse function method: By using the relationship between the definition domain and the value domain of the function f(x) and its inverse function f- 1(x), the value domain of the original function can be obtained by solving the definition domain of the inverse function, and the function value domain with the shape of (a≠0) can be obtained by this method.
(4) Matching method: For the range problem of quadratic function or function related to quadratic function, the matching method can be considered.
(5) Evaluation range of inequality method: Using the basic inequality a+b≥[a, b∈(0, +∞)], we can find the range of some functions, but we should pay attention to the condition of "one positive, two definite, three phases, etc." Sometimes you need skills such as Fang.
(6) Discriminant method: y=f(x) is transformed into a quadratic equation about x, and the definition domain is evaluated by "△≥0". The characteristic of the question type is that the analytical formula contains roots or fractions.
(7) Finding the domain by using the monotonicity of the function: When the monotonicity of the function on its domain (or a subset of the domain) can be determined, the range of the function can be found by using the monotonicity method.
(8) Finding the range of function by combining numbers and shapes: using the geometric meaning expressed by the function, finding the range of function by geometric methods or images, that is, finding the range of function by combining numbers and shapes.
2. Find the difference and connection between the maximum value of the function and the range.
The common method of finding the maximum value of a function is basically the same as the method of finding the function value domain. In fact, if there is a minimum (maximum) number in the range of a function, this number is the minimum (maximum) value of the function. Therefore, the essence of finding the maximum value of a function is the same as that of the evaluation domain, but the angle of asking questions is different, so the way of answering questions is different.
For example, the value range of the function is (0, 16), the maximum value is 16, and there is no minimum value. For example, the range of the function is (-∞, -2]∩[2,+∞), but this function has no maximum and minimum, only after changing the definition of the function, such as X >；; 0, and the minimum value of the function is 2. The influence of definition domain on the range or maximum value of a function can be seen.
3. The application of maximum function in practical problems.
The application of function maximum is mainly reflected in solving practical problems with function knowledge. Many practical problems are commonly expressed in words such as "lowest project cost", "maximum profit" or "maximum (minimum) area (volume)". When solving, we should pay special attention to the restriction of practical significance on independent variables, so as to get the maximum value correctly.
Compulsory 4 Mathematics Chapter 2 Knowledge Points 3 1. A vector can be imagined as a line segment with an arrow. The arrow indicates the direction of the vector; Line segment length: indicates the size of the vector.
2. It is stipulated that if the endpoint A of the line segment AB is the starting point and B is the ending point, the line segment has the direction and length from the starting point A to the ending point B. A line segment with direction and length is called a directed line segment.
3. Modulus of the vector: the size of the vector, that is, the length (or modulus) of the vector. The modulus of vector a is expressed as |a|.
Note: the modulus of the vector is a non-negative real number, and the size can be compared. Because the direction can't compare with the size, and the vector can't compare with the size. The concepts of "greater than" and "less than" are meaningless to vectors.
4. Unit vector: A vector whose length is one unit (i.e. the modulus is 1) is called a unit vector. A vector with a length of 1 in the same direction as the vector A is called the unit vector in the direction A and denoted as a0.
5. A vector with a length of 0 is called a zero vector and recorded as 0. The starting point and ending point of the zero vector coincide, so the zero vector has no definite direction, or the direction of the zero vector is arbitrary.
Calculation of vector
1. Add
Exchange law: a+b = b+a;
Law of association: (a+b)+c=a+(b+c).
minus
If A and B are mutually opposite vectors, then the reciprocal of a=-b, b=-a and a+b=0.0 is 0.
Law of addition and subtraction transformation: a+(-b)=a-b
3. Quantity products
Definition: Two nonzero vectors A and B are known. If OA = A, OB = B, then ∠AOB is called the included angle between vector A and vector B, denoted as θ, and specified as 0≤θ≤π.
Vector product algorithm
A b = b a (commutative law)
(λ a) b = λ (a b) (on the associative law of number multiplication)
(a+b) c = a c+b c (distribution law)
Properties of scalar product of vectors
A a = the square of a |.
a⊥b〈=〉a b=0 .
| a b |≤| a ||| b |. (The formula is proved as follows: | a b | = | a || b||| cos α| Because 0≤|cosα|≤ 1, | AB |≤| A |||||| B |)
What is the way to learn math well in high school?
Mathematics needs to be done with heart, and it is difficult for impetuous people to learn mathematics well. Doing the problem in a down-to-earth manner is the last word.
If you want to learn math well, you can't do it without thinking. You can't hide when you encounter problems, and you can't stop until you figure it out.
The most important thing in mathematics is the process of solving problems. It is important to understand mathematical thinking. With clear thinking, mathematics is naturally learned.
Mathematics is not for looking, but for calculating. Maybe there is nothing in this second, but the second you pick up the pen and start calculating, it will suddenly become clear.
One of the reasons why we can't do math problems is that we haven't studied the examples clearly, so don't let go of the examples in math books.
Odd and even knowledge points of mathematical functions
1. Definition of function parity: For function f(x), if any x in the function definition domain has f(-x)=-f(x) (or f(-x)=f(x)), then function f(x) is called odd function (or even function).
To correctly understand the definitions of odd function and even functions, we should pay attention to two points: the symmetry of the domain on the (1) number axis is a necessary and sufficient condition for the function f(x) to be a odd function or even function; (2)f(x)=-f(x) or f(-x)=f(x) is a unit element in the domain. (Parity is a global property over the domain of a function. ) 。
2. The definition of parity function is the main basis for judging the parity of function. In order to judge the parity of a function, it is sometimes necessary to simplify the equivalent form of the function or application definition.