Interpret parametric lines and zero cross products

For real t and $\mathbf b\ne\mathbf 0$, this is a line through the point with position vector a and parallel to b. If $\mathbf b=\mathbf 0$, it represents the single point a.

When $\mathbf q\ne\mathbf 0$, zero cross product means $\mathbf r-\mathbf p=\lambda\mathbf q$ for some real $\lambda$. Thus

$\mathbf r=\mathbf p+\lambda\mathbf q,\quad\lambda\in\mathbb R.$

R can be any point of the line through P parallel to OQ, including P itself when $\lambda=0$.

If $\mathbf r-\mathbf p=\mathbf 0$, then $R=P$; this is already included above when q is nonzero. If $\mathbf q=\mathbf 0$, the cross product vanishes for every r, so R is unrestricted in three-dimensional space. The given $\mathbf p\ne\mathbf 0$ does not exclude either case.