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.