# linear algebra – Finding equation of line through specified point and perpendicular to the following vector.

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.