Write the normal and vector form of the equation in $\mathbb{R}^2$

Question

This is more of a check then anything else. Here is what I have.

Need to find the normal and vector form of the equation $$-2x+3y=5$$

Normal form: $$(-2,3) \cdot [(x,y) - (-1,1)]$$

Vector form: Now this one I am not sure about. So if someone could care to explain how this is or is not correct I would appreciate it as the textbook is less then helpful so I am just running off my notes.

$$(x,y)= (-1;1) + t(3,2)$$

Answer 1

Your normal form just needs the "equals zero".

As regards your vector form, there is no problem too.

The vector $[(x,y)-(-1,1)]$ must be parallel to vector $(3,2)$ which is perpendicular to $(-2,3)$. So: $$[(x,y)-(-1,1)]=t(3,2) \Rightarrow$$ $$(x,y)= (-1,1)+t(3,2)$$ And we are done.