Q1. How to write the following equation in a form of $\vec{a}= something$?
From the question posted in Physics SE, I found the following process is wrong.
For any $ \vec{v}\ne\vec{0}$ $$ \vec{v}\cdot\vec{a}+c=0 \quad \Rightarrow \quad \vec{v}\cdot(\vec{a}+\frac{\vec{v}}{\vec{v}\cdot\vec{v}}c)=0\quad \Rightarrow \quad \quad \vec{a}+\frac{\vec{v}}{\vec{v}\cdot\vec{v}}c=0\quad \Rightarrow \quad \vec{a}=-\frac{c}{|\vec{v}|^2}\vec{v}$$
According to RogerJBarlow's answer in the above post, "this equation tells you the component of $\vec a$ along $\vec v$, but the component of $\vec a$ perpendicular to $\vec v$ is completely arbitrary."
I want to write the equation in a form of $\vec{a} =something$.
Is it possible? If possible, let me know the method.
Q2. Is this true ?
$$ \vec{v}\cdot\vec{a}+\vec{v}\cdot\vec{c}=0 \quad \Rightarrow \quad \vec{v}\cdot(\vec{a}+\vec{c})=0\quad \Rightarrow \quad \vec{a}=-\vec{c}$$
Thank you.