Vector Equation for a Line
A vector equation for the line $L$ through $P_0(x_0, y_0, z_0)$ parallel to $\mathbf{v}$ is $$\mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{v}, \text{ } -\infty < t < \infty,$$ where $\mathbf{r}$ is the position vector of a point $P(x, y, z)$ on $L$ and $\mathbf{r}_0$ is the position vector of $P_0(x_0, y_0, z_0)$.
Vector Equation for a Plane
A vector equation for the plane through $P_0(x_0, y_0, z_0)$ normal to $\mathbf{n} = A \mathbf{i} + B \mathbf{j} + C\mathbf{k}$ is $$\mathbf{n} \cdot \overrightarrow{P_0P} = 0$$ where $\mathbf{P}$ is any point on the plane.