"Fundamental Theorem of Vectors and Coordinates of Vectors" Preliminary PPT for plane vectors (Fundamental Vector Theorem)

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"Fundamental Theorem of Vectors and Coordinates of Vectors" Preliminary PPT for plane vectors (Fundamental Vector Theorem)

Part One: Explanation of Curriculum Standards

1. Master the basic theorem of collinear vectors and be able to apply it simply.

2. Understand the basic theorem of plane vectors and be able to use a base to represent any vector in the plane.

3. Be able to flexibly apply vector theorem to solve plane geometry problems.

Basic vector theorem and coordinates of vectors PPT, part 2: independent preview before class

1. Basic theorem of collinear vectors

1. Fill in the blanks.

If a≠0 and b∥a, then there is a unique real number λ such that b=λa.

2. How to understand the collinear vector theorem?

Tips: (1) From b=λa⇒a∥b, if λ=0, then b=0, and the zero vector is parallel to any vector. If λ>0, then a and b are in the same direction; if λ<0 , then a and b are opposite.

(2) This theorem has two applications. First, if a vector can be linearly represented by another vector, then the two vectors can be determined to be parallel; second, if two vectors are parallel, then one vector can be linearly represented by another non-zero vector. It can be used to find the parameter λ, which is the basis for vector coordinates on the axis.

3. Do it: if |a|=5, b is opposite to a, and |b|=7, then a=____________b.

2. Basic theorem of plane vectors

1. Fill in the blanks.

Condition: If the two vectors a and b in the plane are not collinear

Conclusion For any vector c in the plane, there is a unique pair of real numbers (x, y) such that c=xa+yb

Base If vectors a and b are not collinear, then {a, b} is called a base representing all vectors in this plane.

2. How to understand the fundamental theorem of plane vectors?

Tips: (1) a and b are two non-collinear vectors in the same plane;

(2) Any vector c in the plane can be linearly represented by a and b, and this representation is unique;

(3) The selection of the base is not unique, as long as two non-collinear vectors in the same plane can be used as a set of bases.

3. Do something: If e1 and e2 are a set of bases in the plane, then the following four sets of vectors can be used as bases of plane vectors ()

A.e1-e2,e2-e1

B.2e1-e2,e1- e2

C.2e2-3e1,6e1-4e2

D.e1+e2,e1-e2

Answer:D

Analysis: e1+e2 and e1-e2 are not collinear and can be used as the basis of plane vectors. The other three sets of vectors are all collinear and cannot be used as the basis.

Basic Vector Theorem and Vector Coordinates PPT, Part 3: Classroom Exploration and Learning

Vector collinearity problem

Example 1 It is known that two non-zero vectors a and b are not collinear, (OA) =a+b, (OB) =a+2b, (OC) =a+3b.

(1) Prove: A, B and C are collinear,

(2) Try to determine the real number k so that ka+b and a+kb are collinear.

Analysis: (1) Prove according to the collinear vector theorem; (2) Use the collinear vector theorem to establish a system of equations to solve.

Application of the Fundamental Theorem of Plane Vectors

Example 2 It is known that in △ABC, D is the midpoint of BC, and E and F are the three equal points of BC. If (AB) =a, (AC) =b, use a and b to represent (AD), ( AE) ,(AF) .

Analysis: Place (AD), (AE), and (AF) in a closed triangle respectively, and use linear operations to continuously move closer to the base.

Solution: From the meaning of the question, we get

(AD)=(AB) +(BD) =(AB) +1/2 (BC) =(AB) +1/2((AC) -(AB) )

=a+1/2(b-a)=1/2a+1/2b,

(AE) =(AB) +(BE) =a+1/3(b-a)=2/3a+1/3b,

(AF) =(AB) +(BF) =a+2/3(b-a)=1/3a+2/3b.

Reflection and understanding: There are mainly two types of vectors used to represent vectors:

(1) Directly use the base, combined with linear operations of vectors, and flexibly apply the triangle rule and the parallelogram rule to solve the problem.

(2) If it is difficult to directly use the base representation, use the principle of "what is difficult is the opposite" and use equation thinking to solve it.

Basic Vector Theorem and Vector Coordinates PPT, Part 4: Thinking Analysis

Application of equation ideas in vectors - mathematical methods

Typical example

As shown in the figure, in ▱ABCD, the midpoints of sides AD and DC are E and F respectively, connecting BE and BF, and intersecting with AC at points R and T respectively. Verify: AR=RT=TC.

From the review perspective, to prove that AR=RT=TC, you only need to find the relationship between AR, AT, and AC. To do this, you need to set parameters with the help of (AR) and (AC) being collinear, and (ER) and (EB) being collinear. Solve the equation to find the parameters.

The basic theorem of vectors and the coordinates of vectors PPT, part 5: detection in class

1. Point C is on line segment AB, and (AC) =3/5 (AB) , (AC) =λ(BC) , then λ is ()

A.2/3 B.3/2 C.-3/2 D.-2/3

Answer:C

2. It is known that a and b are non-collinear vectors, (AB) =λa+2b, (AC) =a+(λ-1)b, and the three points A, B and C are collinear, then λ = ()

A.-1 B.-2 C.-2 or 1 D.-1 or 2

Answer:D

3. In △ABC, E is the midpoint of side AB, F is the midpoint of side AC, and BF intersects CE at point G. If (AG) =x(AE) +y(AF), then xy is equal to ()

A.2/9 B.1/3 C.4/9 D.4/3

Answer:C

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For more information about the "Fundamental Theorem of Plane Vectors and Coordinates of Vectors" PPT courseware, please click on the PPT tag "Fundamental Theorem of Preliminary Vectors and Coordinates of Vectors".

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Update Time: 2024-10-01

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