What will be the vector equation of line in this case?

Question

Given two points, we can definitely find out vector equation of line, but what if I have only a single point and a slope (which is a result of a partial derivative). How do you convert that into a complete vector equation?

Answer 1

How about $p+t\cdot (1,m)$, where $p$ is the point and $m$ the slope...

Answer 2

$\vec r$ = $\vec a$ + n($\vec b$)

where $\vec r$ is the equation of the line, $\vec a$ is the position vector of the point, $\vec b$ is the slope and $n$ is any value in $R$ - {0}

Makes sense?