vectors: are these valid?

If $\displaystyle \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} $ does it follow that $\displaystyle \vec{b} = \vec{c} $?

I said that $\displaystyle \vec{a} \cdot (\vec{b}- \vec{c}) = 0 $ which means that they are perpendicular which implies that $\displaystyle \vec{b} \neq \vec{c} $.

If $\displaystyle \vec{a} \times \vec{b} = \vec{a} \times \vec{c} $ does it follow that $\displaystyle \vec{b} = \vec{c} $?

I said that $\displaystyle \vec{a} \times (\vec{b} - \vec{c}) = 0 $ which means that they are parallel, and so $\displaystyle \vec{b} \neq \vec{c} $.