Orthogonal complement lies on a line with the function $y=ax+b$

Question

Let $v=(10,5)$ be a vector in $\mathbb{R}^2$.

The orthogonal complement to $span(v)$ can be described as the points $(x,y)\in\mathbb{R}^2$ which lies on a line of the function $y=ax+b$.

Determine $a$ and $b$.

How am I supposed to determine this?

Answer 1

Compute $\langle (10,5) , (x,ax+b) \rangle = 5x(2+a)+5b$.

If this equals zero for all $x$ then we must have $b=0, a=-2$.

Answer 2

$(x,y)$ being on the orthogonal complement is defined by $0=(x,y) \cdot (10,5)=10x+5y$.