Usual way of defining Reflection in Plane

We shall call a set $P(a,t)$ a plane in $\hat{\mathbb{R}^n}$(Extended plane) if it is of the form $P(a,t)=\{x\in \mathbb{R}^n: (x.a)=t\}\cup \{\infty\},$ where $a\in\mathbb{R}^n$,$a\ne 0,(x.a)$ usual scalar product and $t$ real . The reflection $\phi$ in $P(a,t)$ is defined in the usual way that is $\phi(x)=x+\lambda a$ where the real parameter $\lambda$ is chosen so that $\frac{1}{2}(x+\phi(x))$ is on $P(a,t).$

My Question is , why it's the usual way to define reflection in plane ? In $\mathbb{R}^2$ if we take $a=(1,0)$ then it's clear . In general intuition is not coming in my mind.