I want to intuitively say tht the answer is yes, but if it so happens that $|\mathbf{a}|=|\mathbf{b}|cos(\theta)$, where $\theta$ is the angle between the two vectors, then the equation will be satisfied without the two vectors being the same.
However, my friend keeps telling me that I'm wrong and that this would contradict the given result in our homework question anyway, which tells us that $\mathbf{a}\times\mathbf{b} = \mathbf{a}-\mathbf{b}$ and then asks us to prove $\mathbf{a}=\mathbf{b}$ (the equation in the question was obtained by dotting both sides with $\mathbf{a}$.
Which one of us is wrong, and why?
Best Answer
As a simple counter-example see
$$\begin{pmatrix}1\\2\\3\end{pmatrix}.\begin{pmatrix}1\\2\\3\end{pmatrix}=14 = \begin{pmatrix}1\\2\\3\end{pmatrix}.\begin{pmatrix}1\\5\\1\end{pmatrix}$$ and clearly
$$\begin{pmatrix}1\\2\\3\end{pmatrix} \not = \begin{pmatrix}1\\5\\1\end{pmatrix}.$$