1. The problem is to find the vector equation of a line. 2. The vector equation of a line passing through a point $\mathbf{r_0}$ with direction vector $\mathbf{d}$ is given by: $$\mathbf{r} = \mathbf{r_0} + t\mathbf{d}$$ where $t$ is a scalar parameter. 3. Here, $\mathbf{r}$ represents the position vector of any point on the line. 4. $\mathbf{r_0}$ is the position vector of a fixed point on the line. 5. $\mathbf{d}$ is the direction vector of the line, indicating its direction. 6. By varying $t$ over all real numbers, the equation describes every point on the line. 7. For example, if the line passes through point $(x_0, y_0, z_0)$ and has direction vector $\langle a, b, c \rangle$, then the vector equation is: $$\mathbf{r} = \langle x_0, y_0, z_0 \rangle + t \langle a, b, c \rangle$$ 8. This can be written in component form as: $$x = x_0 + at, \quad y = y_0 + bt, \quad z = z_0 + ct$$ This fully describes the line in vector form.