If x, , y and z are three unit vectors in three dimensional space… - Vidyadip

MCQ

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

A
$\frac {3}{2}$
✓
$3$
C
$3\sqrt 3$
D
$6$

✓

Answer

Correct option: B.

$3$

b
$(\hat{x}+\hat{y}+\hat{z})^{2} \geq 0$

$\Rightarrow 3+2 \Sigma \hat{x} . \hat{y} \geq 0$

$\Rightarrow 2 \Sigma \hat{x} \hat{y} \geq-3$

Now, $|\hat{x}+\hat{y}|^{2}+|\hat{y}+\hat{z}|^{2}+|\hat{z}+\hat{x}|^{2}$

$=6+2 \Sigma \hat{x} \cdot \hat{y} \geq 6+(-3)$

$\Rightarrow|\hat{x}+\hat{y}|^{2}+|\hat{y}+\hat{z}|^{2}+|\hat{z}+\hat{x}|^{2} \geq 3$

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