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） 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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