I cannot understand this question:
"Find a linear equation (and parametrics) to $v$ where $v$ is perpendicular to the line segment of the extremes $(1,2,1)$ and $B$ $(1,8, -5)$, dividing it in half."
I cannot understand this question:
"Find a linear equation (and parametrics) to $v$ where $v$ is perpendicular to the line segment of the extremes $(1,2,1)$ and $B$ $(1,8, -5)$, dividing it in half."
The question seems to ask you to find the equation of the plane through the midpoint $(1,5,-2)$ of $(1,2,1)$ and $(1,8,-5)$ and which has $(1,2,1)-(1,8,-5)=(0,-6,6)$ as a normal. Hence it is given by $\vec r\cdot(0,-1,1)=(1,5,-2)\cdot(0,-1,1)$ or $y-z=7$.
Parametrically, this is given by
$\{(x,z+7,z):x,z\in\mathbb R\}$
$=\{(0,7,0)+(x,z,z):x,z\in\mathbb R\}$
$=\{(0,7,0)+\lambda(1,0,0)+\mu(0,1,1):\lambda,\mu\in\mathbb R\}$
The midpoint of segment $AB$ is $A*B=(1,5,-2)$ and $(a,b,c)^T=(0,6,-6)^T$ is a direction vector to the line $AB$ so the desired equation of the plane is $$ax+by+cz=d,$$ where $d$ is determined by substituting the coordinates of the midpoint in the equation.