Answer
No.
Counterexample given below.
Work Step by Step
Build a counterexample.
Let ${\bf u}$ =${\bf i, v}$ =${\bf 2i,w}$ =${\bf 3i}., $
The cross products ${\bf u}\times{\bf v}$ and ${\bf u}\times{\bf w}$ both equal ${\bf 0}$, because the cross product of parallel vectors is the zero vector.
So, we have ${\bf u}\times{\bf v}={\bf u}\times{\bf w}, \quad {\bf u}\neq {\bf 0}$,
but ${\bf v}\neq {\bf w} $
