"Plane Vectors and Their Linear Operations" Preliminary PPT courseware for plane vectors (subtraction of vectors)

"Plane Vectors and Their Linear Operations" Preliminary PPT courseware for plane vectors (subtraction of vectors)

Part One: Learning Objectives

Understand the meaning of opposite vectors and be able to use opposite vectors to express the meaning of vector subtraction.

Master the operation of vector subtraction and its geometric meaning, and be able to skillfully perform vector addition and subtraction operations

Can convert vector subtraction operations into vector addition operations

Plane vectors and their linear operations PPT, part 2: independent learning

Problem guide

Preview the contents of textbook P142-P144 and think about the following questions:

1. What is the opposite of a number x? Does a vector a have an opposite vector? If so, how to express it?

2. The sum of any number x and its opposite is 0, then what is the sum of vector a and its opposite?

3. What is the subtraction operation of vectors and its geometric significance?

Plane vectors and their linear operations PPT, Part 3: Preliminary exploration of new knowledge

1. Generally, given any two vectors a and b on the plane, if the vector x can satisfy b + x = a, then x is called the difference between the vectors a and b, and is recorded as ___________. Pick any point O in the plane, draw OA→=a, OB→=b, and draw the vector BA→. Note that OB→+BA→=OA→, so the vector BA→ is the difference between the vectors a and b (also called BA→ is the difference vector between vectors a and b), that is, OA→-OB→=BA→. The above-mentioned graphing method for finding the difference between two vectors is also often called the ____________________ of vector subtraction.

2. Given a vector, we call the vector ____________ and ____________ with this vector its opposite vector, and the opposite vector of vector a is recorded as ______. Because the starting point and end point of the zero vector are the same, -0=______.

It is not difficult to see that a+(-a)=______, AB→+(-AB→)=______.

Vector subtraction can be seen as the inverse operation of vector addition, that is, a-b=____________.

Plane vectors and their linear operations PPT, part 4: self-test

1. Judge whether it is right or wrong (mark “√” if it is correct and “×” if it is wrong)

(1) If b is the opposite vector of a, then a and b must not be equal. ()

(2) If b is the opposite vector of a, then a∥b.()

(3) The opposite vector of vector AB→ is BA→, and BA→=-AB→.()

(4)PA→-PB→=AB→.()

2. The result of simplifying OP→-QP→+PS→+SP→ is equal to ()

A.QP→B.OQ→C.SP→D.SQ→

3. As shown in the figure, in ▱ABCD, AB→=a, AD→=b, using a and b to represent the vectors AC→, BD→, then AC→=________, BD→=________.

4. In parallelogram ABCD, the opposite vector of vector AB→ is ________.

Plane vectors and their linear operations PPT, part 5: interactive lecture and practice

The geometric meaning of vector subtraction

Example 1 As shown in the figure, it is known that the vectors a, b, and c are not collinear. Find the vector a+b-c.

regular method

Two ways to find the difference vector of two vectors

(1) It can be converted into vector addition, such as a-b, you can first do -b, and then do a+(-b).

(2) You can also directly use the triangle rule of vector subtraction, that is, if the starting points of the two vectors coincide, then the difference vector is the vector connecting the end points of the two vectors and pointing to the end point of the vector being subtracted.

Operations and simple applications of vector addition and subtraction

Example 2 (1) Simplification: ①AB→+OA→-OB→=________;

②AB→+(BD→+CA→)+DC→=________;

③OB→-OA→-OC→-CO→=________.

(2) As shown in the figure, ①use a and b to represent DB→;

②Use b and c to represent EC→.

regular method

(1) Common methods for vector subtraction operations

(2) Two forms of vector addition and subtraction simplification

① Connected end to end and form a sum.

②The starting points are the same and different.

When solving problems, pay attention to whether there are these two forms, and pay attention to the reverse application.

(3) Simplification of vector operations related to graphics

First, we must use the arithmetic rules and laws of vector addition and subtraction. Secondly, we must analyze the properties of graphics and assist in simplifying operations through relationships such as vector equality and parallelism in graphics.

Application of geometric significance of vector subtraction

Example 3 It is known that |AB→|=6, |AD→|=9, find the value range of |AB→-AD→|.

[Answer] Because||AB→|-|AD→||≤|AB→-AD→|≤&# 124;AB→|＋|AD→|,

And |AD→|=9, |AB→|=6,

So 3≤|AB→-AD→|≤15.

When AD→ and AB→ are in the same direction, |AB→-AD→|=3;

When AD→ and AB→ are opposite, |AB→-AD→|=15.

Therefore, the value range of |AB→-AD→| is [3, 15].

regular method

(1) Steps to solve plane geometry problems using vector method

①Abstract quantities in plane geometry problems into vectors.

②Convert the problem into a vector problem and perform vector operations.

③Restore vector problems to plane geometry problems.

(2) The method and key to prove that a quadrilateral is a parallelogram using the vector method

①Use vectors to prove that the line segments are parallel and equal, thereby proving that the quadrilateral is a parallelogram. You only need to prove that the vectors represented by the corresponding directed line segments are equal.

②Flexibly apply vector operation rules based on graphics and find the relationship between vectors is the key to solving such problems.

Plane vectors and their linear operations PPT, Part 6: Feedback on compliance with standards

1. In parallelogram ABCD, AC→-AD→ is equal to ()

A.AB→B.BA→

C.CD→ D.DB→

2. The following equation:

①0-a＝-a; ②-(-a)＝a; ③a＋(-a)＝0; ④a＋0＝a; ⑤a-b＝a＋(-b); ⑥a＋(-a)＝0.

The correct number is ()

A. 3B. 4

C． 5 D. 6

3. Simplify BA→-CA→+DB→-DC→=________.

4. It is known that OA→=a, OB→=b, if |OA→|=5, |OB→|=12, and ∠AOB=90°, then | ;a-b|=________．

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