EUCLIDEAN LINE, PLANE AND SPACES #EUCLIDEAN #LINE #PLANE #SPACES

EUCLIDEAN LINE, PLANE AND SPACES     

 We know about the real line or real number system R, R is one dimensional.

Consider the two dimensional plane R²={(a ,b)/ a ,b Ꜫ R}.

In R² ,the elements of the form (a ,b) are called the ordered pairs of a & b.

In R², the addition and multiplication are defined as

for x=(a₁,b₁) , y=(a₂,b₂) in R²                               
                                      

x +y=(a₁₊a₂,b₁₊b₂ ) and x .y=(a_1.a₂ ,b_1.b₂)

Here R² is called as an Euclidean  plane.

Now consider the three dimensional space R³={(a ,b ,c )/ a ,b ,c Ꜫ R}.

In R³,the elements of the form (a ,b ,c) are called ordered triples of a, b, c.

The addition and multiplication in  R³ are same as in R² .

Similarly in R⁴, the elements of the form (a ,b ,c ,d) called as ordered 4- tuples of real numbers in which the addition and multiplication are same as in R² &  R³.

∴In general the n^th dimensional Euclidean space Rⁿ of all ordered n-tuples of real numbers of the form (a₁,a_2,…,a_n )

i.e.,  Rⁿ = {(a₁,a_2,…,a_n ) / a₁,a_2,…,a_n Ꜫ R}.                             

 In Rⁿ the addition and multiplication are defined as 

for x= (a₁,a_2,…,a_n ) ; y=( b₁,  b₂,…,  b_n) Ꜫ Rⁿ, where a_i , b_j ꜪR 

x+y= (a₁+  b₁, a₂+ b₂,…, a_n+ b_n) and 

x.y = (a₁.  b₁, a₂. b₂,…, a_n. b_n)  also

for any 𝛂 Ꜫ R,   ax= (𝛂a₁, 𝛂a_2,…, 𝛂a_n ).

In    Rⁿ  , the absolute value of x is defined as |x| = √ (a₁²+a₂²+…+a_n² )  

                                                            = square root (a₁²+a₂²+…+a_n² )    

          * for n>2, the sets Rⁿ are called Euclidean spaces,

          * R² is called Euclidean plane whereas 

          * R is called an Euclidean line.

#euclidean   #space #line #plane

#pair #ordered #n-tuple #addition #multiplication