Let $a_1^2+a_2^2+a_3^2=1, b_1^2+b_2^2+b_3^2=4 \& c_1^2+c_2^2+c_3^2=9$, then the maximum value of $\l

Question

Let $a_1^2+a_2^2+a_3^2=1, b_1^2+b_2^2+b_3^2=4 \& c_1^2+c_2^2+c_3^2=9$, then the maximum value of $\left(a_1 b_2 c_3+a_2 b_3 c_1+a_3 b_1 c_2\right)-\left(a_1 b_3 c_2+a_2 b_1 c_3+a_1 b_2 c_3\right)$ is

Professor PC Thomas Classes · Answer

Let $\vec{a}=a_1 i+a_2 j+a_3 k, \vec{b}=b_1 j+b_2 j+b_3 k \quad \& \vec{c}=c_1 i+c_2 j+c_3 k$, then $\left(a_1 b_2 c_3+a_2 b_3 c_1+a_3 b_1 c_2ight)$
$
-\left(a_1 b_3 c_2+a_2 b_1 c_3+a_3 b_2 c_1ight)=\left|\begin{array}{lll}
a_1 & b_1 & c_1 \
a_2 & b_2 & c_2 \
a_3 & b_3 & c_3
\end{array}ight|=[\vec{a} \vec{b} \vec{c}]
$
Now $[\vec{a} \vec{b} \vec{c}]<|\vec{a}||\vec{b}||\vec{c}|$
Hence
$\left|\begin{array}{lll}a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3\end{array}ight| \leqslant$\ $\sqrt{a_1^2+a_2^2+a_3^2} \sqrt{b_1^2+b_2^2+b_3^2} \sqrt{c_1^2+c_2^2+c_3^2}$
$\Rightarrow\left|\begin{array}{lll}a_1 & b_1 & c_1 \ a_2 & b_2 & c_2 \ a_3 & b_3 & c_3\end{array}ight| \leqslant 6$