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What problems should be paid attention to in high school mathematics vector?
Vector part

1. Plane vector knowledge structure table

2. The concept of vector

The Basic Concept of (1) Vector

(1) The quantity that defines the size and direction is called a vector. The size of the vector is also the length of the vector, which is called the module of the vector.

(2) A vector with a specific size or a specific relationship.

Zero vector, unit vector, * * * line vector (parallel vector), equal vector, opposite vector.

③ Representation: Geometric method: Draw a directed line segment to represent it, and record it as or α.

(4) In the coordinate system, any point on the plane can be represented by a pair of real numbers (coordinates). Based on two unit vectors on the X-axis and Y-axis, a vector on the plane =x +y is denoted as =(x, y), which is called the coordinate of the vector.

=(x2-x 1, y2-y 1), where a (x 1, y 1) and b (x2, y2).

(2) Vector operation

① Addition and subtraction of vectors: definitions and rules (as shown in Figure 5-1);

a+b=(x 1+x2,y 1+y2),a-b=(x 1-x2,y 1-y2)。 Where a=(x 1, y 1) and b = (x2, y2).

Algorithm: a+b = b+a, (a+b)+c = a+(b+c), A+0 = 0+A = A.

(2) the number of vector multiplication (product of real number and vector) definition and law (as shown in figure 5-2):

λa=λ(x,y)=(λx,λy)

( 1)︱ ︱=︱ ︱? 6? 1︱ ︱;

(2) When > 0, it is in the same direction; When < 0, the opposite;

When =0, = 0.

(3) If = (), then? 6? 1 =( ).

Arithmetic law

λ(μa)=(λμ)a,(λ+μ)a=λa+μa,λ(a+b)= λa+λb .

3. The definition and law of the product of plane vectors (as shown in Figure 5-3):

(1). Vector included angle: given two non-zero vectors and b, if =, =, then ∠AOB= () is called the included angle of vector sum.

(2). Quantity product of two vectors:

Given the sum of two nonzero vectors and their included angle is, then

? 6? 1 =︱ ︱? 6? 1︱ ︱cos。

Where cos is called the projection of the vector in the direction.

(3) The nature of the product of vectors:? 6? 1 = ? 6? 1 ,(λ )? 6? 1 = ? 6? 1(λ )=λ( ? 6? 1 ),( + )? 6? 1 = ? 6? 1 + ? 6? 1 。 If =(),=()? 6? 1 =

ⅰ) ⊥ ? 6? 1 =0 (,non-zero vector);

Ii) The vector and the included angle are acute angles.

Iii) The vector and the included angle are obtuse.

4. Theorems and formulas

① * * * straight line theorem: The necessary and sufficient condition for the straight line between vector B and non-zero vector A * * is that there is only one real number λ, so that B = λ a..

Conclusion: ∨ (? 8? The necessary and sufficient condition of 2) is x 1y2-x2y 1=0.

Note: 1? 8? 3 λ cannot be divided by two formulas, ∫y 1, y2 may be 0, ∫? 8? ∴ At least one of x2, y2 and y2 is not 0.

2? 8? 3 The necessary and sufficient condition cannot be written as ∵ x 1, and x2 may be 0.

3? 8? 3 vector * * * line has two necessary and sufficient conditions: ∨ (? 8? 2 )

② Basic quantization of plane vectors: If there are two nonlinear vectors on the same plane, then any vector on this plane has only a pair of real numbers λ 1, and λ2 makes =λ 1 +λ2.

③ Necessary and sufficient conditions for two vectors to be perpendicular.

㈠⊥? 6? 1 = 0㈡⊥x 1? 6? 1x2+y 1? 6? 1y2=0( =(x 1,y 1),=(x2,y2))

④ Three-point * * * line theorem: The necessary and sufficient condition of three-point A, B and C*** lines on the plane is that there are real numbers α and β, so that = α+β, where α+β= 1, and O is any point on the plane.

⑤ Numerical calculation formula

The formula of the distance between two points is || =, where [p 1 (x 1, y 1), p2 (x2, y2)]

The ratio of p-divided directed line segments:

Let P 1 and P2 be two points on a straight line, and point P is any point in the world different from P 1 and P2, then there is a real number that makes =, which is called the ratio of point P to directed line segment.

When point p is on the line segment, > 0; When point P is on the extension line of line segment or, < 0;

Formula of vernal equinox coordinates: if =;; The coordinates of are (), () and () respectively; Then; Midpoint coordinate formula:

The included angle formula of two vectors: cosθ= =

0≤θ≤ 180,a=(x 1,y 1),b=(x2,y2)

⑥ Graphic transformation formula Translation formula: If point P0(x, y) is translated to P(x', y') according to vector a=(h, k),

rule

⑦ Relevant conclusions

(i) There are any three points o, a and b on the plane. (+) If m is the midpoint of the line segment AB;

Generally speaking, if p is the bisector of the line segment AB and the ratio is constant λ (i.e. = λ, λ≦- 1), then =+is the vector formula of the bisector of the line segment.

(ii) Add a finite number of vectors, a 1, a2, …, an. You can start from point O and make vectors =a 1, =a2, …, =an one by one, then the vector is the sum of these vectors, that is,

A 1+a2+…+an=++…+ = (polygon rule of vector addition).

When An and o coincide (that is, when the above-mentioned polyline OA 65438+OA 2...an is a closed polyline), the sum vector is zero.

Note: It is an important means to solve the vector problem by using the sum formula of the above vectors, that is, expressing a vector as the sum of several vectors.

3. The application of vectors

Application of (1) vector in geometry (2) Application of vector in physics.