minwaardekn

2022-06-24

I'm reading a textbook at the moment that provides the following linear equation,
$\alpha \mathbf{v}+\mathbf{v}×\mathbf{a}=\mathbf{b},$
and asks to solve for $\mathbf{v}$. The form of $\mathbf{v}$ is given as
$\mathbf{v}=\frac{{\alpha }^{2}\mathbf{b}-\alpha \left(\mathbf{b}×\mathbf{a}\right)+\left(\mathbf{a}\cdot \mathbf{b}\right)\mathbf{a}}{\alpha \left({\alpha }^{2}+|\mathbf{a}{|}^{2}\right)}.$
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 $\alpha \ne 0$.

Colin Moran

Expert

Taking cross product with $\mathbf{a}$ on both sides, we get,

Now solve for $\mathbf{v}$ directly.

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