Answer
No.
Counterexample given below.
Work Step by Step
Counterexample: Let
${\bf u}=\langle 1,1\rangle$
${\bf v_{1}}=\langle 1,0\rangle,\quad {\bf v_{2}}=\langle 0,1\rangle$
${\bf u}\cdot{\bf v_{1}}=1+0=1,$
${\bf u}\cdot{\bf v_{2}}=0+1=1.$
${\bf u}\cdot{\bf v_{1}}={\bf u}\cdot{\bf v_{2}}$, but ${\bf v_{1}}\neq{\bf v_{2}}$
