How to learn vector related problems well in compulsory mathematics in senior one.

Unit 5 plane vector

Lecture on the concept and operation of 17 plane vector

Key points of knowledge

Some Concepts of 1. Plane Vector

Definition of (1) vector: (2) Representation method: use directed line segments to represent the vector, indicate the size of the vector, and use the direction indicated by the arrow to represent _ _ _ _ _. Use letters, … or …

(3) Modulus: The length of the vector is called || or ||.

(4) Zero vector: The vector of is called zero vector, which is recorded as; The direction of the zero vector.

(5) Unit vector: called unit vector.

(6)*** line vectors: vectors with the same or opposite directions are called

Also called * * * line vector, it specifies any vector of _ _ _ _ _ and * * * lines.

(7) Equal vector: Equal vector.

(8) Inverse quantity: sum is called inverse quantity;

2. Vector operation

Addition of (1) vectors:

Definition: The operation of finding the sum of two vectors is called.

2 rules: rules; Parallelogram rule.

③ Operation law: exchange law; Association law;

(2) Vector subtraction:

Definition: The operation of finding the difference between two vectors is called.

Rule 2: triangle rule; Law.

(3) the arrow points;

3. Product of real number and vector:

(1) Definition: The product of a real number and a vector is a vector, remember.

Work, specifying: || = |||. When > 0, the direction of and the direction of; When < 0, the direction of and the direction of; When =0, is parallel to.

(2) Algorithm: () =,

( + ) = , ( + )= .

4. Two important theorems:

(1) Vector * * * Straight Line Theorem: Between Vector and Nonzero Vector * * *

The necessary and sufficient condition is that there is only one real number, so that it =, that is.

(2) Basic theorem of plane vectors: If there are two vectors on the same plane, there is only one pair of real numbers for any vector on this plane, so;

5. Quantity product of vectors:

Angle of (1) vector: As shown below, the sum of two non-zero vectors is known.

, as =, =, then ∠AOB= () is called the included angle between vector and, and is recorded as.

(2) Definition of product of quantity: given the sum of two non-zero vectors, and the included angle is, then this quantity is called the product of the sum quantity, and it is recorded as? , namely:? =| || | .

(3) The geometric meaning of product of quantity: product of quantity? The modulus is equal to the product of the projection in the sum direction.

(4) The nature of the product of quantities: Let it be a unit vector, =.

① ? = ? = .

(2) When the directions are the same? = ; When the direction is opposite. =, especially, =, or || =. ③ ⊥ _ _ _.④ = _ _ _ _ _ _ _.⑤|? |≤| || |.

(5) Algorithm: ①? = ? ; ②( )? = = ( ); ③( + )? = .

Case analysis

Linear operation of a question vector

Example 1 As shown in the figure, the two diagonals of a parallelogram intersect at one point, and,, are represented by,, and respectively.

Analysis: In the parallelogram,

, ,

The Application of Two Vector Lines of Question Types

In Example 2, it is assumed that there are two known non-linear vectors.

, ,

If three points, * * * line, find the value.

Analysis: Because there are * * * lines, any two vectors composed of three points can be regarded as * * * lines, and the value of k can be obtained by using the necessary and sufficient condition column relationship of vector * * * lines.

,, * * ok,,

* * * line, existence makes, that is

, ,

Three-plane vector norm problem of question type

Example 3 shows that the angle between and is, and is: (1); (2) .

Analysis: (1),

, ,

(2)

Example 4 shows that they are all non-zero vectors, and they are consistent with.

Vertical, vertical, find the angle with.

Analysis: From the known,

That is to say,

, two types also want to reduce.

Substitute it into any of these formulas,

, therefore.

homework

1. If it is not a zero vector, it is satisfied.

The angle between and is ()

A.B. C. D。

2. In the middle, point on it and divide it equally. If,,, then ()

A.B.

C.D.

3. If the known vector satisfies, then ()

A. 0 BC to 4 BC

4. Given,,, the included angle with is ().

A.B. C. D。

5. Known,,, then the projection of the vector in the vector direction is ()

A.B.4 C. D.2

6. Assuming that the included angle between the vector sum is 0,0, then _ _ _ _ _ _

7. Let a vector satisfy:,,, then _ _ _ _ _ _ _

8. Yes, if. If this is satisfied, then _ _ _ _ _ _

9. If the point is the midpoint of the line segment and the point is outside the straight line, then _ _ _ _.

10. Set two non-zero vectors and non-* * lines.

(1) If,

, verification:,, three-point * * * line;

(2) Try to determine the real number to make it conform to * * *.

1 1. The known vector,, satisfies,

⊥ ⊥. If, ask

The value.

12. It is a parallelogram with a vector side.

Lecture 18 Basic Determination of Plane Vector

Coordinate representation of 1. plane vector

In rectangular coordinate system, make arbitrary vectors based on the same axis and axis direction respectively. According to the basic theorem of plane vector, there is only one pair of real numbers, so we can mark the so-called rectangular coordinates as coordinates on the X axis and coordinates on the Y axis. The so-called coordinates are coordinates with equal pointing quantities. Obviously,

2. Coordinate operation of plane vector

(1) is known, then _ _ _ _ _ _ _ _ _ _ _

(2) If the real number is known, then _ _ _

(3) If,, then ‖ is a necessary and sufficient condition; If so, then _ _ _ _ _ _ _

(4) If you click, then _ _ _

(5) If, then _ _ _

(6) If the starting point coordinates and ending point coordinates of the vector are respectively,

, then _ _ _ _ _; This is the distance formula between two points;

(7) set, then _ _ _ _ _ _ _;

(8) =_____________

3. The demarcation point of the line segment

(1) Let it be two points on a straight line, which is any point different from this point in the world, then there is a real number, which is called the ratio of the divided and directed line segments and the fixed point. When, on the extension line of, call;

When, on the extension line of, it is called the outer point; When, between and the midpoint; When, for the midpoint; When, between the midpoint and; When coincidence, it is called;

(2) The ratio of points is _ _ _ _ (any point on the plane)

(3) The coordinate formula of the fixed fraction point is,

, _______, ________;

When it is the midpoint, _ _ _ _ _, _ _ _ _.

4. Graphic translation

(1) is set as a graph in the coordinate plane, and all points on the graph move in the same direction by the same length to get a graph. This process is called graphic translation. When a graph is translated, its position in the coordinate plane will change.

(2) Translation formula: Let it be any point on f, if the corresponding point on the graph of vector translation is, then there is; If you know one of the three points, you can get the other point, but pay attention to the order.

Case analysis

Coordinate representation and operation of problem type vector

Given three vectors in the 1 plane,

(1) Find a satisfactory real number;

(2) If ‖, be realistic;

(3) If ‖, and ask.

Analysis: (1) Judging from the meaning of the question,

So, you have to.

(2) , ,

, .

(3) Suppose that,,, comes from the meaning of the problem, or solved, or.

Problem 2 uses the coordinates of plane vectors to solve the problem of vector straight lines.