let vector a=i+j+k and let r be a variable vector such that r.i , r.jand r.k are positive integers. if r.a<=12 then what are the no. of values of r?

Dear shefali

 let r =xi +yj +zk

r.i =x ≥1

r.j=y  ≥1

r.k=z  ≥1

and r.a =x+y+z ≤12

  x+y+z ≤12

let u=x-1

   v=y-1

  w=z-1

 then   u+v+w ≤9     where u≥0,v≥0,w≥0

now we have to find the intiger solution of this equation.

let  u+v+w =n   where n varies from 0 to 9

  solution of these equation

 n:0 to 9  ∑^n+3-1C_3-1

=n:0 to 9  ∑ⁿ⁺²C₂

=n:0 to 9  ∑(n+2)(n+1)/2

=n:0 to 9  1/2 ∑(n²+3n+2)

= 1/2 [9(9+1)(18+1)/6   + 3*9(9+1)/2  + 2 *10]

 =1/2[285 +135+20] =220

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