Interpret a plane equation and derive the line-intersection parameter

(a) Points P,Q,R have position vectors $\mathbf p,\mathbf q,\mathbf r$. P and Q are fixed; R varies. Given $\mathbf r=(x,y,z)$, $\mathbf p=(-1,2,4)$, $\mathbf q=(3,-5,2)$ and $(\mathbf r-\mathbf p)\cdot\mathbf q=0$, find the relation between x,y,z and describe the locus of R.

(b) Interpret $\mathbf r\cdot\mathbf n=d$ geometrically, where $\mathbf n$ is a constant unit vector and d is a constant scalar. State what d represents.

(c) Given $\mathbf b\cdot\mathbf n\ne0$, solve $\mathbf r=\mathbf a+t\mathbf b$ and $\mathbf r\cdot\mathbf n=d$ for t in terms of $\mathbf a,\mathbf b,\mathbf n,d$. Interpret the solution geometrically.