Question

Prove that|(a ⃗+(b)) ⃗×(a ⃗-(b)) ⃗ | = 2ab. If a ⃗⊥ b ⃗.

Answer 1

Answer on Question #44083, Math, Vector Calculus

Prove that $|(a^{-1} + (b))^{-1} \times (a^{-1} - (b))^{-1}| = 2ab$. If $a^{-1} \perp b^{-1}$.

**Solution.**

If $\vec{a} \perp \vec{b}$, then $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}| \sin 90{}^\circ = |\vec{a}||\vec{b}|$.

$$\left| (\vec {a} + \vec {b}) \times (\vec {a} - \vec {b}) \right| = | \vec {a} \times \vec {a} - \vec {a} \times \vec {b} + \vec {b} \times \vec {a} - \vec {b} \times \vec {b} | = | 0 - \vec {a} \times \vec {b} - \vec {a} \times \vec {b} - 0 | = 2 | \vec {a} \times \vec {b} | =$$ $$2 | \vec {a} | | \vec {b} |.$$

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