Answer
This implies that the vector ${\bf{v}} - {\bf{w}}$ is perpendicular to the vector ${\bf{a}}$. However, it is not true that ${\bf{v}}$ is always equal to ${\bf{w}}$.
Work Step by Step
Let ${\bf{v}}$, ${\bf{w}}$, and ${\bf{a}}$ be nonzero vectors such that ${\bf{v}}\cdot{\bf{a}} = {\bf{w}}\cdot{\bf{a}}$. So,
${\bf{v}}\cdot{\bf{a}} - {\bf{w}}\cdot{\bf{a}} = 0$
$\left( {{\bf{v}} - {\bf{w}}} \right)\cdot{\bf{a}} = 0$
This implies that the vector ${\bf{v}} - {\bf{w}}$ is perpendicular to the vector ${\bf{a}}$. However, it is not true that ${\bf{v}}$ is always equal to ${\bf{w}}$.
