Is this vector parallel to the plane So I am given two points $A=(-.5,2.3,-7.3)$ and $B=(-2,17.1,-0.3)$ and then using $AB = OB - OA$ to give me $(1.5,-14.8,-7)$. The plane is $$x+23y+13z=500$$ From there I got $r.n$ where $r=(1.5,-14.8,-7)$ and $n=(1,28,13)$. From here I do not know how to check if the vector is perpendicular to the plane.
We have the two points $A(-0.5,2.3,-7.3)$ and $B(-2,17.1,-0.3)$, as well as the general equation of the plane $x+23y+13z-500=0$. The vector from $A$ to $B$ can easily be found.
$$\vec{\mathbf{r}}=\langle(-2+0.5),(17.1-2.3),(-0.3+7.3)\rangle=\langle-1.5,14.8,7\rangle$$
The normal vector to the plane is $\vec{\mathbf{n}}=\langle1,23,13\rangle$. These two vectors are parallel if their cross product is equal to $\vec{\mathbf{0}}$ [1]. If $\vec{\mathbf{r}}$ and $\vec{\mathbf{n}}$ are parallel vectors, then they are both perpendicular to the plane in question.
$$\vec{\mathbf{r}}\times\vec{\mathbf{n}}=\langle(14.8)(13)-(7)(23),(7)(1)-(-1.5)(13),(-1.5)(23)-(14.8)(1)\rangle$$
This comes from the fact that $\vec{\mathbf{u}}\times\vec{\mathbf{w}}=\langle u_2w_3-u_3w_2,u_3w_1-u_1w_3,u_1w_2-u_2w_1 \rangle$ [2].
$$\vec{\mathbf{r}}\times\vec{\mathbf{n}}=\langle31.4,26.5,-49.3\rangle\ne\langle0,0,0\rangle$$
Let us see if the vector $\vec{\mathbf{r}}$ happens to be parallel to the plane in question. If it is, then the dot product of $\vec{\mathbf{r}}$ and $\vec{\mathbf{n}}$ will be $0$ as the two vectors will be orthogonal.
$$\vec{\mathbf{r}}\cdot\vec{\mathbf{n}}=(-1.5)(1)+(14.8)(23)+(7)(13)=-1.5+340.4+91=429.9\ne0$$
The vector from $A$ to $B$ is not perpendicular to the plane in question, nor is it parallel to it.
In general two vectors: $(a_1, a_2, a_3)$ and $(b_1, b_2, b_3)$ are said to be parallel if
$$\frac{a_1}{b_1}=\frac{a_2}{b_2}=\frac{a_3}{b_3}$$
Now, the vector $\vec{OB}=(-1.5,14.8,7)$ will be perpendicular to the plane:$x+23y+13z=500$ iff  vector $r=(-1.5,14.8,7)$ is parallel to the normal vector $n=(1,28,13)$ of plane i.e.
$$\frac{-1.5}{1}\ne\frac{14.8}{28}\ne \frac{7}{13}$$
above inequality shows that vector $r=(-1.5,14.8,7)$ is not parallel to the normal vector $n=(1,28,13)$ i.e. vector $r=(-1.5,14.8,7)$ is not perpendicular to the given plane.