What is the 'meaning' behind $r·n = a·n$

Question

I was confused about what the dot product represents and I think I have grasped that now from this post What does the dot product of two vectors represent? .

However, I still cannot understand what $r·n = a·n$ represents in the same way. I am not sure if this is the universal notation for it, but I was told that '$r$' is any point on a plane $(x,y,z)$ and '$n$' is the normal to the plane with '$a$' being any known point on the plane. I can't understand why the dot product of the normal and ($O$ - any point on the plane) would be the same as the dot product of the normal and ($O$ - one particular point on the plane) as ($O$ - a point on the plane) would be a different vector every time where n stays constant surely resulting in a different dot product.

Probably haven't articulated my misunderstanding too well but any help appreciated.

Answer 1

Another way to visualize the equation is that if you construct an alternative orthonormal basis in $\mathbb R^3,$ with basis vectors $\mathbf e_1,$ $\mathbf e_2,$ and $\mathbf e_3,$ you can get the coordinates of any vector $\mathbf r$ over the alternative basis by taking the dot product of $\mathbf r$ with each basis vector. For example, the first coordinate is $\mathbf r \cdot \mathbf e_1.$

Given a unit vector $\mathbf n$ normal to some plane, you can make it part of an orthonormal basis by setting $\mathbf e_1=\mathbf n.$ The other two basis vectors then are parallel to the plane. If you take an arbitrary vector $\mathbf a$ and then take all vectors $\mathbf r$ such that $\mathbf r \cdot \mathbf n= \mathbf a \cdot \mathbf n,$ those vectors describe a plane through $\mathbf a$ parallel to your initial plane in the same way that all vectors with first coordinate $k$ describe a plane parallel to the plane $x=1.$ If $\mathbf a$ is the position vector of a point in your initial plane then the “parallel” plane is simply the plane you started with.

To prove that this all works, you would probably need to do some algebraic manipulations. But once you have proved that it works, you can visualize $\mathbf r\cdot \mathbf n$ as one of the coordinates of $\mathbf r$ in an orthonormal basis with $\mathbf n$ as one of the basis vectors without thinking again about the algebra you used in the proof.

Answer 2

If you rearrange it, it is $$(\mathbf{r} - \mathbf{a})\cdot\mathbf{n} = 0.$$ Since $\mathbf{r} - \mathbf{a}$ is the vector joining $\mathbf{a}$ to $\mathbf{r}$, this equation says that $\mathbf{n}$ is orthogonal (perpendicular) to the vector joining $\mathbf{a}$ to $\mathbf{r}$ for any $\mathbf{r}$ on the plane. If you draw a diagram, you should be able to intuitively see that this is the case.