挖掘单位向量的内涵,利用单位向量的特性解题(Excavate the connotation of unit vector and solve the problem with unit vector characteristic)
挖掘单位向量的内涵,利用单位向量的特性解题(Excavate the
connotation of unit vector and solve the problem with unit
vector characteristic)
Data worth having
From the usual study, accumulation and summary
Where there is a problem, there must be some
Please also criticize and correct me!
Mining the connotation of unit vectors
Solving problems with the characteristics of unit vectors
Zhang Wenyao, senior middle school, Guzhen Town, Guangdong, Zhongshan
[Abstract] a unit vector is a vector whose modulus is equal to 1 unit lengths
This is for high school teachers and students
Very familiar
But have you ever tried to dig out the unique properties of unit vectors?
What is the use of these characteristics of unit vectors to solve problems? Even by using these properties, we can
skillfully and simply solve some of the more complex mathematical problems Here are some concrete examples to illustrate how to mine the meaning of unit vectors How can I use some of its special properties to solve problems? I hope we can give you some inspiration for your future math study
[Keywords] the unit vectors, the modules of vectors, the scalar product of vectors
[text] what are the special properties of a unit vector? How do we use these features of vectors to solve mathematical problems skillfully? We might as well use letters to represent unit vectors
By digging into the meaning of unit vectors
The characteristics of the unit vector are summarized as follows:
1, unit vector definition: length equal to 1 units of vector, called unit vector (unit vector), people's version of compulsory 4
2, the expression of unit vector (that is the common form of unit vector):
= = (1)
0) = = (0)
1) = = (
);......
If a nonzero vector is set
Die for
Then the unit vector can also be represented as:
4.
(not at the same time zero)
3, the special properties of unit vectors:
By digging
We simply sum up the properties of a unit vector:
Two equal vectors of units in the same direction
The modular length of a unit vector is equal to 1
That is
Let's study: how to use these two special properties in solving
mathematical problems of it
One or two vectors of equal units in the same direction
Example 1, known point A (1)
-2)
Vectors and = (2)
3) same direction
= find the coordinates of the point B
The conventional solution is to set the coordinates of the point
B
Again by vectors and = (2
3) same direction (that is collinear)
According to the condition of collinearity of vectors and the system of equations of =
Solution equations are obtained
Get the coordinates of B
Let's call this solution a solution
Other solutions are denoted as solutions two,......
Here's a detailed look at the problem solving process
Solution of a point B;
be
By C.
have to
That is
Solution equations are obtained
or
or
or
Inspection
To do
therefore
The coordinates of point B are (5)
4)
Without exception, everyone will find that the solution of two yuan and two equations is quite complicated and time-consuming
After solving the system of equations, it needs to be checked
To find the right answer
Therefore, we hope to have a concise solution
The following solution, two, is the use of vector properties
to solve the problem
Method: two points B
be
= dreams
And to the same unit vector r
And dreams (= 2
3)
And to the same unit vector r
And dreams by A.
Is the unit vector of two identical directions
Should be equal
L
Be solved
Coordinate point for B (5 *
4)
Compare the above two different solutions
What have you found? What is the harvest?
Example 2, if plane vector and = (1)
-2) the included angle is
And
Coordinate of vector
The conventional solution is no longer necessary
Here we mainly discuss how to solve the problem by using the
properties of unit vectors
Solution:
And
And to the same unit vector r =
Dreams (= 1
-2)
And to the same unit vector r
And dreams by A.
Vectors and = (1)
-2) the included angle is
L and are two opposite unit vector
L
Solution
Coordinate vector for R
Comparing the two solutions
It's easy to find out
Using the nature of a vector two equal vectors, there is a unique
way in the operation of the solution vector
It can greatly simplify the calculation and simplify the generalization
To solve the problem of the objective
Two, the modular length of a unit vector is equal to 1
That is
Example 3, (people's Education Edition compulsory 4) seeks the maximum and minimum functions
Analysis: in this case, the structure on the right side of the function equals the coordinate operations of the two vectors and the scalar product
Therefore, the method of construction is adopted
Think of a function as two vectors = = 4
3) the scalar product of unit vectors
That is, then, by using the properties of vector scalar product and when and same direction
When and reverse
The maximum and minimum values of the function can be obtained
The concrete solution is as follows:
Solution: set = (4)
3),
Then,
Dreams
L by vector properties
have to
When same as the same direction
Y has maximum value
;
When and reverse
Y has minimum values
We can also use the properties of vectors equal to the vectors of two identical units
Find the angle at which the function gets the maximum and minimum (no longer here)
You may as well try
Use the method described above to find the maximum values of the following trigonometric functions and the corresponding angles
1, to find the maximum and minimum function
At a corresponding angle
2, (people's Education Edition compulsory 4) with a, B for the
maximum and minimum values
by force of contrast
You'll find solutions to the properties of unit vectors
It can simplify and simplify the process
I hope this will help you with your study
Improve your thinking and problem solving skills
There will be some revelations for you
Hope more exchanges
Discuss together
Common progress
common development
In May 12, 2008, it was published in Guzhen high school
Reference:
"Ordinary high school curriculum standards experimental textbooks (mathematics required 4)" (people's Education Edition), November 2007 third printing
Middle school second teaching materials (high school
mathematics compulsory 4) (supporting people's Education Edition experimental textbook), Beijing Education Press, 06, March Edition
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