Let V be the volume of the parallelopied formed by the vectors
`overset(to)(a) =a_(1) hat(i) +a_(2)hat(j) +a_(3)hat(k) , overset(to)(b) =b_(1)hat(i) +b_(2)hat(j) +b_(3)hat(k)`
`" and " overset(to)(C) =c_(1)hat(i) +c_(2)hat(j) +c_(3)hat(k)`
If `a_(r) , b_(r) , c_(r)` where r=1,2,3 are non-negative real numbers and `overset(3)underset(r=1)(Sigma) (a_(r)+b_(r)+c_(r))=3L.` Show that `V le L^(3)`

Explanation

`V= |vec(a).(vec(b)xx vec(c ))|le sqrt(a_(1)^(2)+a_(2)^(2)+a_(3)^(2))`
`sqrt(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))sqrt(c_(1)^(2)+c_(2)^(2) +c_(3)^(2))`
Now `L= ((a_(1) +a_(2)+a_(3))+(b_(1)+b_(2)+b_(3))+(c_(1)+c_(2)+c_(3)))/(3)`
`gt [(a_(1)+a_(2)+a_(3))(b_(1)+b_(2)+b_(3))(c_(1)+c_(2)+c_(3))]^(1//3)`
`rArr L^(3) ge [(a_(1)+a_(2)+a_(3))(b_(1)+b_(2)+b_(3))(c_(1)+c_(2)+c_(3))] ....(ii)`
Now `(a_(1) +a_(2) +a_(3))^(2)`
`rArr (a_(1) + a_(2)+a_(3)) gt sqrt(a_(1)^(2)+a_(2)^(2) +a_(3)^(2))`
Similarly `(b_(1)+b_(2)+b_(3)) ge sqrt(b_(1)^(2)+b_(2)^(2)+b_(3)^(2))`
and `(c_(1) +c_(2) +c_(3) ) ge sqrt(c_(1)^(2) +c_(2)^(2)+c_(3)^(2))`
`:. L^(3) ge [(a_(1)^(2) +a_(2)^(2) +a_(3)^(2)) (b_(1)^(2)+ b_(2)^(2)+b_(3)^(2))(c_(1)^(2)+c_(2)^(2)+c_(3)^(2))]^(1//2)`
`rArr L^(3) ge V `[from Eq.(i))]