Relation between parameters of a vector field and it's projection

[center o bass]

Say we have two vector fields X and Y and we form the projection of Y, Y' orthogonal to X. Since every vector field is associated with a curve with a corresponding parameter, is there a relation between the parameters of Y and Y'?

 

[fzero]

The projection of Y orthogonal to X can be expressed as (a scalar multiple of)

$$Y' = Y - \frac{\langle X, Y\rangle X }{ \langle X, X \rangle} ,$$

wherever $\langle X, X \rangle\neq 0$. We can define the flow curves via $\dot{\gamma}_Y(t) = Y \gamma(t)$ and $\dot{\gamma}_{Y'}(t) = Y' \gamma(t)$. In principle we can use the same parameter $t$ to describe both of these curves, but the curve for $Y'$ will also depend on the vector field $X$, so I don't think there's a general answer to your question. If $X$ and $Y$ are given and nice enough, then we can just solve for the curves to work out the relationship.