Let $\alpha,\beta\in\mathbb R$, and $W\in\mathbb R^n$. Consider $$\alpha V+\beta \frac{V}{\lVert V\rVert}=W$$ where $\lVert \cdot\rVert$ is the regular Euclidean norm. Is there a closed form solution for unknown $V\in\mathbb R^n$?
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$\begingroup$ Note that $W$ and $V$ are parallel, so reduce to an equation in $\mathbb{R}$. $\endgroup$– user10354138Commented Oct 9, 2018 at 18:42
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$\begingroup$ That should do the trick. Works nicely, thanks! $\endgroup$– PeterACommented Oct 9, 2018 at 18:48
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Following up on @user10354138, comment $V$ and $W$ are parallel, hence there exists a constant $k$ such that $V=kW$. Hence $$ \alpha kW+\beta \frac{kW}{\lVert kW\rVert}=W \;\Leftrightarrow\; \alpha kW+\beta \;\text{sign}(k)\frac{W}{\lVert W\rVert}=W, $$ and therefore the problem is reduced to $\mathbb R$ $$ \alpha k+\beta \;\text{sign}(k)\frac{1}{\lVert W\rVert}=1, $$ hence $$ k = -\text{sign}(k)\frac{\beta}{\alpha\lVert W\rVert}\;. $$