ΕΣΩΤΕΡΙΚΟ ΓΙΝΟΜΕΝΟ ΚΑΙ ΑΠΟΔΕΙΚΤΙΚΕΣ ΑΣΚΗΣΕΙΣ

ΕΣΩΤΕΡΙΚΟ ΓΙΝΟΜΕΝΟ ΚΑΙ ΑΠΟΔΕΙΚΤΙΚΕΣ ΑΣΚΗΣΕΙΣ

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ΛΥΣΗ

α) Έχουμε:

$\lvert \vec{u} + \vec{\nu} \rvert^{2} + \lvert \vec{u} - \vec{\nu} \rvert^{2} =$

$(\vec{u} + \vec{\nu})^{2} + (\vec{u} - \vec{\nu})^{2} =$

$\vec{u}^{2} + 2 \vec{u} \cdot \vec{\nu} + \vec{\nu}^{2} + \vec{u}^{2} - 2 \vec{u} \cdot \vec{\nu} + \vec{\nu}^{2} =$

$2 \vec{u}^{2} + 2 \vec{\nu}^{2} =$

$2 \lvert \vec{u} \rvert^{2} + 2 \lvert \vec{\nu} \rvert^{2}.$

β) Έχουμε:

$\frac{1}{4} \lvert \vec{u} + \vec{\nu} \rvert^{2} - \frac{1}{4} \lvert \vec{u} - \vec{\nu} \rvert^{2}=$

$\frac{1}{4} (\vec{u} + \vec{\nu})^{2} - \frac{1}{4} (\vec{u} - \vec{\nu})^{2} =$

$\frac{1}{4} (\vec{u}^{2} + 2 \vec{u} \cdot \vec{\nu} + \vec{\nu}^{2}) - \frac{1}{4} (\vec{u}^{2} - 2 \vec{u} \cdot \vec{\nu} + \vec{\nu}^{2}) =$

$\frac{1}{4} \vec{u}^{2} + \frac{1}{4}\cdot 2 \vec{u} \cdot \vec{\nu} + \frac{1}{4} \vec{\nu}^{2} - \frac{1}{4} \vec{u}^{2} + \frac{1}{4} \cdot 2 \vec{u} \cdot \vec{\nu} - \frac{1}{4} \vec{\nu}^{2} =$

$\frac{1}{4} \vec{u}^{2} + \frac{1}{2} \vec{u} \cdot \vec{\nu} + \frac{1}{4} \vec{\nu}^{2} - \frac{1}{4} \vec{u}^{2} + \frac{1}{2} \vec{u} \cdot \vec{\nu} - \frac{1}{4} \vec{\nu}^{2} =$