"Plane Vectors and Their Linear Operations" Preliminary PPT of plane vectors (multiplying vectors, linear operations of vectors)

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"Plane Vectors and Their Linear Operations" Preliminary PPT of plane vectors (multiplying vectors, linear operations of vectors)

Part One: Explanation of Curriculum Standards

1. Understand the concept of vector multiplication and understand the geometric meaning of multiplication operations.

2. Understand and master the operational laws of vector multiplication, and be able to perform vector multiplication operations.

3. Understand and master the properties and determination methods of two vectors being collinear, and be able to skillfully use this knowledge to deal with issues related to vector collinearity.

4. Be able to use vector addition, subtraction and multiplication to perform linear operations.

Plane vectors and their linear operations PPT, part 2 content: independent preview before class

1. Multiply vectors

1. Fill in the blanks.

Define the operation of multiplying a real number λ by a vector a

Notation λa

Length|λa|=|λ||a|

direction

The direction of λ>0 is the same as the direction of a

The direction of λ<0 is opposite to the direction of a

2.What is the geometric meaning of multiplication of numbers?

Tip: The geometric meaning of the multiplication of vectors is to expand or reduce the vector a along the direction of a or the opposite direction of a. When λ>0, it expands (λ>1) or shrinks (0<λ) along the direction of a. <1)λ times; when λ<0, expand (|λ|>1) or shrink (|λ|<1)|λ& in the opposite direction of a #124; times.

3. Do it: think and analyze

(1) For any vector a, there is always 0·a=0.()

(2) When λ>0, |λa|=λa.()

(3) If a≠0, λ≠0, then the directions of a and -λa are opposite. ()

Answer: (1)× (2)× (3)×

Analysis: (1) 0·a=0; (2)|λa|=λ|a|(λ>0). (3) When λ<0, -λ>0 , a has the same direction as -λa.

2. Multiply vectors and linear arithmetic laws

1. Fill in the blanks.

(1)λ(μa)=(λμ)a;

(2)(λ+μ)a=λa+μa;

(3)λ(a+b)=λa+λb.

2. How to understand the operational law of vector multiplication?

Tips: (1) The law of multiplication of vector numbers is very similar to the law of multiplication of real numbers. However, due to different factors, the distributive law of vector multiplication can be divided into (λ+μ)a=λa+μa and λ(a+b)= λa+λb.

(2) The theoretical basis of the vector multiplication law is the definition that two vectors are equal. Therefore, the key to proving this law is to prove that the modules of the vectors on both sides of the equation are equal and have the same direction. And make a comprehensive study of various possible situations. discussion.

Plane vectors and their linear operations PPT, part three: classroom exploration and learning

Linear operations on vectors

Example 1 Simplify the following expressions:

(1)2(5a-4b+c)-3(a-3b+c)-7a;

(3)(m+n)(a-b)-(m+n)(a+b).

Analysis: Just simplify according to the addition, subtraction and multiplication operations of vectors.

Solution: (1) Original formula =10a-8b+2c-3a+9b-3c-7a=b-c.

(3) Original formula=(m+n)a-(m+n)b-(m+n)a-(m+n)b=-2(m+n)b.

Summary of methods for reflection and understanding of vector multiplication operations

(1) The multiplication operation of vectors is similar to the algebraic operation of polynomials. The transformation methods such as removing brackets, shifting terms, merging similar terms, and extracting common factors in real number operations are also applicable to the product of numbers and vectors, but here " "Like terms" and "common factors" point to vectors, and real numbers are regarded as coefficients of vectors.

(2) Vectors can also be solved through a series of equations. Treat the vector you want as an unknown number and use the method of solving algebraic equations. At the same time, pay more attention to observation during the operation process, appropriately apply the laws of operation, and simplify the operation.

Plane vectors and their linear operations PPT, part 4: thinking analysis

Vector collinearity problem - mathematical method

Typical example: It is known that the non-zero vectors e1 and e2 are not collinear. If (AB) =e1+2e2, (BC) =-5e1+6e2, (CD) =7e1-2e2, then the three collinear points are___________ .

Analysis: Convert the three-point collinear problem into a vector collinear problem. For example, (AB) ∥(BD) can deduce that A, B, and D are collinear.

Analysis: ∵(AB) =e1+2e2,(BD) =(BC) +(CD) =-5e1+6e2+7e1-2e2

=2(e1+2e2)=2(AB) ,

∴(AB) and (BD) are collinear and have a common point B.

∴The three points A, B, and D are collinear.

Answer:A,B,D

Variant training knows that m and n are non-collinear vectors, a=3m+4n, b=6m-8n, and determine whether a and b are collinear.

Solution: If a and b are collinear, then there exists λ∈R such that a=λb,

That is, 3m+4n=λ(6m-8n).

∵m,n are not collinear,∴{■(6λ=3"," @"-" 8λ=4"." )┤

∵There is no λ that satisfies this system of equations at the same time, and ∴a and b are not collinear.

Plane vectors and their linear operations PPT, Part 5: Inspection in class

1. It is known that point D is a point on the plane where △ABC is located, satisfying (BD) =1/4 (DC), then (AD) = ()

A.1/4 (AB) +3/4 (AC) B.3/4 (AB) +1/4 (AC)

C.4/5 (AB) +1/5 (AC) D.1/5 (AB) +4/5 (AC)

2. It is known that △ABC and point M satisfy (MA) + (MB) + (MC) =0. If there is a real number m such that (AB) + (AC) =m(AM) holds, then the value of m is __________ .

3. As shown in the figure, in △ABC, (AD) =2/3 (AC), (BP) =1/3 (PD), if (AP) =λ(AB) +μ(AC), then λ+ The value of μ is ()

A.11/12 B.3/4 C.8/9 D.7/9

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For more information about the PPT courseware "Plane Vectors Preliminary Plane Vectors and Their Linear Operations Multiplied by Linear Operations of Vector Vectors", please click on the Preliminary PPT Plane Vectors Plane Vectors and Their Linear Operations PPT Multiplying Vectors PPT Linear Operations of Vectors PPT tag.

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Update Time: 2024-10-03

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