Vectors 3 Question 28
28. Let $V$ be the volume of the parallelopiped formed by the vectors
$\mathbf{c}=c _1 \mathbf{i}+c _2 \hat{\mathbf{j}}+c _3 \mathbf{k}$
If $a _r, b _r, c _r$, where $r=1,2,3$ are non-negative real numbers and $\sum _{r=1}^{3}\left(a _r+b _r+c _r\right)=3 L$. Show that $V \leq L^{3}$. $(2002,5 M))$
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Solution:
- $\quad V=|\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})| \leq \sqrt{a _1^{2}+a _2^{2}+a _3^{2}}$
$$ \sqrt{b _1^{2}+b _2^{2}+b _3^{2}} \sqrt{c _1^{2}+c _2^{2}+c _3^{2}} $$
Now, $L=\frac{\left(a _1+a _2+a _3\right)+\left(b _1+b _2+b _3\right)+\left(c _1+c _2+c _3\right)}{3}$
$$ \geq\left[\left(a _1+a _2+a _3\right)\left(b _1+b _2+b _3\right)\left(c _1+c _2+c _3\right)\right]^{1 / 3} $$
[using AM $\geq$ GM]
$\Rightarrow L^{3} \geq\left[\left(a _1+a _2+a _3\right)\left(b _1+b _2+b _3\right)\left(c _1+c _2+c _3\right)\right]$
Now, $\left(a _1+a _2+a _3\right)^{2}$
$=a _1^{2}+a _2^{2}+a _3^{2}+2 a _1 a _2+2 a _1 a _3+2 a _2 a _3 \geq a _1^{2}+a _2^{2}+a _3^{2}$
$\Rightarrow \quad\left(a _1+a _2+a _3\right) \geq \sqrt{a _1^{2}+a _2^{2}+a _3^{2}}$
Similarly, $\left(b _1+b _2+b _3\right) \geq \sqrt{b _1^{2}+b _2^{2}+b _3^{2}}$
and $\quad\left(c _1+c _2+c _3\right) \geq \sqrt{c _1^{2}+c _2^{2}+c _3^{2}}$
$\therefore \quad L^{3} \geq\left[\left(a _1^{2}+a _2^{2}+a _3^{2}\right)\left(b _1^{2}+b _2^{2}+b _3^{2}\right)\left(c _1^{2}+c _2^{2}+c _3^{2}\right)\right]^{1 / 2}$
$\Rightarrow \quad L^{3} \geq V$
[from Eq. (i)]
