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The simplest problem of analytic geometry

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a2
Consider the Cartesian rectangular coordinate system in the plane. Taking the projection of any point М1 on the x and y coordinatewe obtain two numbers x=a1 and y=b1. Take two numbers a2 and b2 plot a2 on the x -axis and b2 on the y -axis. Having drawn two straight lines parallel to the coordinate axes through these points we find obtain a point M 2 in their intersection.

M2(a2;b2)
y

M1(a1;b1)

b2

       
   
 
 


0

a1

    Thus, there is a one–to–one correspondence between points in the plane and pairs of numbers.

Analytic geometry (all of its statements, theorems, and formulas) can be constructed on the basis elementary school mathematics. But we use tools of vector algebra in derivations and proofs.

The distance between two points. Let us find the distance between two points М 1 and М 2 in the plane.

y

 

d M2(x2;,y2)

M1(x1;y1)

0 x

 

 

Compose the vector .

The length of this vector is defined by

 

.

 

This is the distance between the two given points.

Division of an interval in a given ratio. Suppose given an interval М 1 М 2. Let us find the coordinates a of point М on the interval for which .

Compose the vectors and .

y

 

M2(x2,y2)

M(x,y)

M1(x1,y1)

0 x

 

It is known from vector algebra that the condition for two vectors to be collinear is the proportionality of their respective coordinates:

.

From the first fraction we obtain

; ; .

This gives the x coordinate; y is found in a similar way:

; .

To obtain a formula for the midpoint of the interval, we take l =1:

; .


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