Question

Let $x^2+y^2+z^2=5,x,y,z\in R,$ then the maximum value of $(x-y{)}^2+(y-z{)}^2+(z-x{)}^2$ is

Answer 1

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$
$\overrightarrow{b}=x\hat{i}+y\hat{j}+z\hat{k}$
$\overrightarrow{a}\times \overrightarrow{b}=(z-y)\hat{i}+(x-z)\hat{j}+(y-x)\hat{k}=|\overrightarrow{a}||\overrightarrow{b}|\sin \theta$
$\Rightarrow (z-y{)}^2+(x-z{)}^2+(y-x{)}^2=3\times 5{\sin }^2\theta$
So, the maximum value is $15$