<s>[INST] Given the equation $\alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b}$, solve for $\mathbf{v}$. I'm reading a textbook at the moment that provides the following linear equation,
$$
\alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b},
$$
and asks to solve for $\mathbf{v}$. The form of $\mathbf{v}$ is given as
$$
\mathbf{v} = \frac{\alpha^2 \mathbf{b} - \alpha (\mathbf{b} \times \mathbf{a}) + (\mathbf{a}\cdot\mathbf{b})\mathbf{a}}{\alpha(\alpha^2+\lvert \mathbf{a} \rvert^2)}.
$$
It's easy enough to verify that this is the correct solution. However, I can't figure out how I'd solve for $\mathbf{v}$ if given just the original equation.
Are there any general approaches to solving this kind of equation systematically?
Edit: $\mathbf{a}, \mathbf{b}$ and $\mathbf{v}$ are all vectors, whereas $\alpha$ is a scalar such that $\alpha \neq 0$.
 [/INST] Taking cross product with $\mathbf{a}$ on both sides, we get,
\begin{align*}
&\alpha \mathbf{v} + \mathbf{v}\times \mathbf{a} = \mathbf{b}\\
\implies &\alpha(\mathbf{v}\times \mathbf{a})+(\mathbf{v}\times \mathbf{a})\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\\
\implies &\alpha(\mathbf{b}-\alpha \mathbf{v})+(\mathbf{v}\cdot \mathbf{a})\mathbf{a}-|a|^2\mathbf{v}=\mathbf{b}\times \mathbf{a}\\
\implies &\alpha \mathbf{b}-\alpha^2\mathbf{v}+\dfrac1\alpha (\mathbf{b}\cdot \mathbf{a})\mathbf{a}-|a|^2\mathbf{v}=\mathbf{b}\times \mathbf{a}&&\Big(\text{Using }\alpha (\mathbf{v}\cdot \mathbf{a})=\mathbf{b}\cdot \mathbf{a}\Big)
\end{align*}
Now solve for $\mathbf{v}$ directly.
</s>