Chứng minh 2 vecto cùng phương, 2 vecto cùng hướng (hay, chi tiết)

Kiến Thức Y Khoa 25/08/2023

0 4 minutes read

Table of Contents

Introduction

In this article, we will delve into the topic of proving the parallelism and direction of vectors. By providing a detailed step-by-step approach, this article aims to help students revise and understand how to solve problems related to proving the parallelism and direction of vectors effectively.

A. Solving Method

Definition:

The line passing through the initial and terminal points of a vector is called its line of action.
Two vectors are said to be parallel if their lines of action are either parallel or coincide.
Two parallel vectors can have the same or opposite directions.
Convention: The zero vector (represented by 0) is considered parallel and in the same direction as any vector.

Three vectors are said to be parallel to each other:

Vector a is parallel to vector b and vector c if the lines of action of all three vectors are parallel.

Solution Method:

To prove that two vectors are parallel, we need to demonstrate that their lines of action are parallel or coincide. This can be done through various mathematical techniques, such as using the properties of perpendicular lines, parallel lines, the Thales theorem, the median properties of triangles, trapezoids, and the congruent alternate angles theorem.

To prove that two vectors have the same direction, we need to show that they are parallel and then examine their respective directions.

B. Illustrative Examples

Example 1:

Consider a regular hexagon ABCDEF centered at O. How many vectors, other than zero, are parallel to the vectors connecting the vertices of the hexagon?

Solution:
Since ABCDEF is a regular hexagon with center O, we can deduce that:

BE || CD || AF
Hence, OB || CD || AF

Therefore, the vectors parallel to the vectors connecting the vertices of the hexagon are those with initial and terminal points as the vertices of the hexagon.

Hence, the answer is 6.

Example 2:

Given two vectors, a and b, that are not parallel. Which of the following statements is correct?

A. There are no vectors parallel to both a and b.
B. There are infinitely many vectors parallel to both a and b.
C. There is one vector parallel to both a and b, and it is vector c.
D. All of the above statements are incorrect.

Solution:

As per the convention, any vector is parallel and in the same direction as itself.
Hence, statement C is correct, implying that there exists at least one vector, c, which is parallel to both a and b.
Statements A and D are incorrect because the existence of vector c is proof that there are vectors parallel to both a and b.

Therefore, the correct answer is C.

Example 3:

Given point A and vector b different from vector a, determine the point M such that vector m is parallel to vector a.

Solution:
Let the line of action of vector a be the line l.

Case 1: Point A lies on line l

We can choose any point M on line l.
In this case, vector m = am is parallel to vector a.

Case 2: Point A does not lie on line l

Draw a line m parallel to line l passing through point A.
Choose any point M on line m.
In this case, vector m = am is parallel to vector a.

Hence, the point M lies on a line passing through point A and is parallel to line l.

Example 4:

Which of the following statements is true?

A. Two vectors are parallel to a third vector if they have the same direction.
B. Two vectors are parallel to a third vector if they have the same direction and the third vector is different from them.
C. Two vectors are parallel to a third vector if they have the same direction irrespective of the third vector.
D. Two vectors are parallel to a third vector if they have opposite directions.

Solution:
A. False, as two vectors can have the same direction but have different third vectors.
B. False, as the third vector can be the same as the given vectors, and they may not be parallel to each other.
C. True, as parallel vectors have the same direction regardless of the third vector.
D. False, as two vectors with opposite directions can be parallel to a third vector.

Hence, the correct answer is C.

Example 5:

Given three distinct points A, B, and C. Which statement is most accurate?

A. Points A, B, and C are collinear only if they are parallel.
B. Points A, B, and C are collinear only if they are parallel and in the same direction.
C. Points A, B, and C are collinear only if they are parallel and have the same direction.
D. Points A, B, and C are collinear regardless of whether they are parallel or not.

Solution:

Points A, B, and C are collinear only if they are parallel is incorrect, as points can be collinear without being parallel.
Points A, B, and C are collinear only if they are parallel and in the same direction is incorrect, as parallelism and direction are not the only conditions for collinearity.
Points A, B, and C are collinear only if they are parallel and have the same direction is incorrect, as parallelism and direction are not sufficient conditions for collinearity.

Hence, the only remaining option is D, which states that points A, B, and C are collinear regardless of whether they are parallel or not.

Conclusion

This article aimed to provide a detailed explanation of how to prove the parallelism and direction of vectors. By following the step-by-step approach presented in the article, students can enhance their understanding and problem-solving skills in this topic. Remember, practice makes perfect! For more practice and detailed solutions to various vector-related problems, visit Kienthucykhoa.com.

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