If $|(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a})| = 12$, where $\vec{a}$ is a unit vector, find $|\vec{x}|$.

$\sqrt{13}$

The correct answer is Option (2) → $\sqrt{13}$ ##

Since $\vec{a}$ is a unit vector, $|\vec{a}| = 1$.

$ (\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 12$

$⇒\vec{x}\cdot\vec{x} + \vec{x}\cdot\vec{a} - \vec{a}\cdot\vec{x} - \vec{a}\cdot\vec{a} = 12$

$⇒|\vec{x}|^2 - |\vec{a}|^2 = 12$

$⇒|\vec{x}|^2 - 1 = 12$

$⇒|\vec{x}|^2 = 13 ⇒ |\vec{x}| = \sqrt{13}$