I have three vectors: $\overrightarrow{u}$ ,$\overrightarrow{v}$ and $\overrightarrow{w}$ and this equality: $\overrightarrow{u}\xb7\overrightarrow{v}=\overrightarrow{u}\xb7\overrightarrow{w}$. Then the question is if $\overrightarrow{v}=\overrightarrow{w}$. At first i tried to write the equality representing components:

$\overrightarrow{u}\xb7\overrightarrow{v}={u}_{1}\xb7{v}_{1}+\dots +{u}_{n}\xb7{v}_{n}$

$\overrightarrow{u}\xb7\overrightarrow{w}={u}_{1}\xb7{w}_{1}+\dots +{u}_{n}\xb7{w}_{n}$

Then:

${u}_{1}\xb7{v}_{1}+\dots +{u}_{n}\xb7{v}_{n}={u}_{1}\xb7{w}_{1}+\dots +{u}_{n}\xb7{w}_{n}.$

But I got stuck right there; I thought if ${c}_{i},i\in [1,n]$ could be canceled from both terms, but don't know if that is possible somehow.

$\overrightarrow{u}\xb7\overrightarrow{v}={u}_{1}\xb7{v}_{1}+\dots +{u}_{n}\xb7{v}_{n}$

$\overrightarrow{u}\xb7\overrightarrow{w}={u}_{1}\xb7{w}_{1}+\dots +{u}_{n}\xb7{w}_{n}$

Then:

${u}_{1}\xb7{v}_{1}+\dots +{u}_{n}\xb7{v}_{n}={u}_{1}\xb7{w}_{1}+\dots +{u}_{n}\xb7{w}_{n}.$

But I got stuck right there; I thought if ${c}_{i},i\in [1,n]$ could be canceled from both terms, but don't know if that is possible somehow.