Question

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors, then the maximum value of ${|2\overrightarrow{a}-3\overrightarrow{b}|}^2+{|2\overrightarrow{b}-3\overrightarrow{c}|}^2+{|2\overrightarrow{c}-3\overrightarrow{a}|}^2$ is

Answer 1

Let $A(1,y_1)$ and $B(x_2,11)$ be two points on the curve $y=x^2-2x+3$.
If $O$ is the origin, then $\overrightarrow{OA}.\overrightarrow{OB}$
$y=x^2-2x+3$
$\Rightarrow y_1=2$ when $x=1$ and $x^2-2x+3=11$
$\Rightarrow x^2-2x-8=0$
$\Rightarrow x=4,-2$ are the roots
$\therefore x_2=4$ or $-2$
$A=(1,2),B=(4,11)$ or $(-2,11)$
$\overrightarrow{OA}.\overrightarrow{OB}=(\hat{i}+2\hat{j}).(4\hat{i}+11\hat{j})=26$
(or) $(\hat{i}+2\hat{j})(-2\hat{i}+11\hat{j})=-2+22=20$