IIT-JEE 2002Question 1
Let $V$ be the volume of the parallelopiped formed by the vectors $\vec a=a_1\hat i+a_2\hat j+a_3\hat k$, $\vec b=b_1\hat i+b_2\hat j+b_3\hat k$, $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$, where $r=1,2,3$, are non-negative real numbers and $\displaystyle\sum_{r=1}^{3}(a_r+b_r+c_r)=3L$, show that $V\le L^{3}$.
Vector Algebra
Answerproved
Solution
$V=|\det|\le|\vec a||\vec b||\vec c|$ (Hadamard). For non-negative entries $|\vec a|\le a_1+a_2+a_3$ etc. So $V\le(\sum a_r)(\sum b_r)(\sum c_r)\le\left(\frac{\sum(a_r+b_r+c_r)}3\right)^3=L^3$ by AM-GM.