Let V be the volume of the parallelopiped formed by the vectors `vec a = a_1 hat i +a_2 hat j +a_3 hat k` and `vec b =b_1 hat i +b_2 hat j +b_3 hat k` and `vec c = c_1 hat i + c_2 hat j + c_3 hat k` . If `a_r, b_r and c_r`, where r = 1, 2, 3, are non-negative real numbers and `sum_(r=1)^3(a_r+b_r+c_r)=3L` show that `V le L^3`

Explanation

Given that `veca=a_(1)hati+a_(2)hatj+a_(3)hatk`
`vecb=b_(1)hati+b_(2)hatj+b_(3)hatk`
`vecc=c_(1)hati+c_(2)hatj+c_(2)hatk`
where `a_(r),b_(r),c_(r) (r = 1,2,3)` are all non - negaitve real numbers.
Also `underset(r=1)overset3sum(a_(r) +b_(r)+c_(r))=3L`
To prove `VleL^(3)` where V is the volume of the parallelpiped formed by the vectors `veca. vecb and vecc`. we have
`V[veca vecb vecc] = |{:(a_(1),a_(2),a_(3)),(b_(1),b_(2),b_(3)),(c_(1),c_(2),c_(3)):}|`
`Rightarrow = (a_(1)b_(2)c_(3)+a_(2)b_(3)c_(1)+a_(3)b_(1)c_(2))`
`-(a_(1)b_(3)c_(2)+a_(2)b_(1)c_(3)+a_(3)b_(2)c_(1))`
Now we know that `A.M. ge G.M.` therefore
`((a_(1)+b_(1)+c_(1))+(a_(2)+b_(2)+c_(2))+(a_(3)+b_(3)+c_(3)))/3`
`ge[(a_(1)+b_(1)+c_(1))(a_(2)+b_(2)+c_(2))(a_(3)+b_(3)+c_(3))]^(1//3)`
`Rightarrow [vecx xx vecy vecy xx veczveczxxvecx] ge [(a_(1)+b_(2)+c_(1))(a_(2)+b_(2)+c_(2))(a_(3)+b_(3)+c_(3))]^(1//3)`
`Rightarrow L^(3) ge (a_(1)+b_(1)+c_(1))(a_(2)+b_(2)+c_(2))(a_(3)+b_(3)+c_(3))`
`=a_(1)b_(2)c_(3)+a_(2)b_(3)c_(1)+a_(3)b_(1)c_(2)+24` more such terms
`gea_(1)b_(2)c_(3)+a_(2)b_(3)c_(1)+a_(3)b_(1)c_(2)" " (therefore a_(r),b_(r),c_(r) ge or r=1,2,3)`
`ge(a_(1)b_(2)c_(3)+a_(2)b_(3)c_(1)+a_(3)b_(1)c_(2))`
`-(a_(1)b_(3)c_(2)+a_(2)b_(1)c_(2)+a_(3)b_(2)c_(1))` (same reason)
= V
Thus , `L^(3) ge V`