Matrix/Vector equation involving the norm of the vector

Question

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 $V\in\mathbb R^n$?

Answer 1

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 $$ k = -\text{sign}(k)\frac{\beta}{\alpha\lVert W\rVert}\;. $$