If $\vec{a}$ is a unit vector and $(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8$, then find $|\vec{x}|$.

3

The correct answer is Option (2) → 3 ##

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

$(\vec{x} - \vec{a}) \cdot (\vec{x} + \vec{a}) = 8$

or $\vec{x} \cdot \vec{x} + \vec{x} \cdot \vec{a} - \vec{a} \cdot \vec{x} - \vec{a} \cdot \vec{a} = 8$

or $|\vec{x}|^2 - 1 = 8 \text{ i.e. } |\vec{x}|^2 = 9$

Therefore $|\vec{x}| = 3$ (as magnitude of a vector is non negative).