I have the following question. Suppose we have the point $P=(1,4,-3)$ and the vector $v=langle -1,4,0 rangle.$ I want to find the equation of the line that passes through the point $P$ that is also perpendicular to $v$. So of course letting $Q=(x,y,z)$ be a point on this line tells us that $langle x-1,y-4,z+3rangle cdot langle-1,4,0 rangle = 0$. This tells us that $-x+4y=15$. My question is how do we retrieve the parametric, vector, and symmetric form of the line from this equation?

Thanks again.

Krull.