If $\hat{x}, \hat{y}$ and $\hat{z}$ are three unit vectors in threedimensional space, then the minimum value of $|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2$

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Aicharya · Accepted Answer

$(\hat{x}+\hat{y}+\hat{z})^2 \geq 0$ $ \begin{aligned} &\Rightarrow 3+2 \Sigma \hat{x} \cdot \hat{y} \geq 0 \ &\Rightarrow 2 \Sigma \hat{x} \cdot \hat{y} \geq-3 \end{aligned} $ Now, $|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2$ $=6+2 \sum \hat{x} . \hat{y} \geq 6+(-3)$ $ \Rightarrow|\hat{x}+\hat{y}|^2+|\hat{y}+\hat{z}|^2+|\hat{z}+\hat{x}|^2 \geq 3 $

If $\hat{x}, \hat{y}$ and $\hat{z}$ are three unit vectors in threedimensional space, then the minimum value…

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