If x, y and z are three unit vectors in three-dimensional space, then the minimum value of…

Question

If $\hat x,\hat y$ and $\hat z$ are three unit vectors in three-dimensional space, then the minimum value of $|\hat x+\hat y|^2+|\hat y+\hat z|^2+|\hat z+\hat x|^2$ is:

Accepted Answer

$|\hat x+\hat y+\hat z|^2\ge0 \implies 3+2(\hat x\cdot\hat y+\hat y\cdot\hat z+\hat z\cdot\hat x)\ge0 \implies \hat x\cdot\hat y+\hat y\cdot\hat z+\hat z\cdot\hat x\ge-\dfrac32$

$$|\hat x+\hat y|^2+|\hat y+\hat z|^2+|\hat z+\hat x|^2 = 6+2(\hat x\cdot\hat y+\hat y\cdot\hat z+\hat z\cdot\hat x) \ge 6-3=3$$

Solutions · 1