Prove that ( vector a + vector b) * ( vector a - vector b) = a ^2 - b ^2?

Aug 2, 2018

Please see a Proof in Explanation.

Explanation:

Recall that the dot product of vectors is distributive and

commutative.

$\therefore \left(\vec{a} + \vec{b}\right) \cdot \left(\vec{a} - \vec{b}\right)$,

$= \vec{a} \cdot \left(\vec{a} - \vec{b}\right) + \vec{b} \cdot \left(\vec{a} - \vec{b}\right)$,

$= \vec{a} \cdot \vec{a} - \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{b}$

$= {\vec{a}}^{2} - \cancel{\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{b}} - {\vec{b}}^{2}$, as desired!