Let V be the volume of the parallelepied formed by the vectors, `veca = a_(1)hati=a_(2)hatj + a_(3) hatk , vecb = b_(1) hati + b_(2)hatj + b_(3) hatk and vecc =c_(1)hati + c_(2)hatj + c_(3)hatk . if a_(r) b_(r) nad 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 leL^(3)`

Explanation

To solve the problem, we need to show that the volume \( V \) of the parallelepiped formed by the vectors \( \vec{a}, \vec{b}, \vec{c} \) is less than or equal to \( L^3 \) given the condition that \( \sum_{r=1}^{3} (a_r + b_r + c_r) = 3L \).

### Step-by-Step Solution:

1. **Volume of the Parallelepiped**:
The volume \( V \) of the parallelepiped formed by the vectors \( \vec{a}, \vec{b}, \vec{c} \) can be expressed as:
\[
V = |\vec{a} \cdot (\vec{b} \times \vec{c})|
...