1. The problem is to understand the vector equation of a line and how to represent it visually. 2. The vector equation of a line passing through a point $\mathbf{a}$ with direction vector $\mathbf{d}$ is given by: $$\mathbf{r}(t) = \mathbf{a} + t\mathbf{d}$$ where $t$ is a scalar parameter. 3. Here, $\mathbf{a}$ is the position vector of a fixed point on the line, and $\mathbf{d}$ indicates the direction of the line. 4. To draw the diagram: - Start by plotting the point corresponding to $\mathbf{a}$. - From this point, draw the vector $\mathbf{d}$. - The line consists of all points $\mathbf{r}(t)$ obtained by scaling $\mathbf{d}$ by $t$ and adding it to $\mathbf{a}$. 5. This means as $t$ varies over all real numbers, $\mathbf{r}(t)$ traces the entire line. 6. The key rule is that the direction vector $\mathbf{d}$ must not be the zero vector, otherwise the "line" reduces to a single point. 7. In summary, the vector equation provides a parametric way to describe every point on the line using a fixed point and a direction vector. Final answer: The vector equation of a line is $$\mathbf{r}(t) = \mathbf{a} + t\mathbf{d}$$ where $\mathbf{a}$ is a point on the line and $\mathbf{d}$ is the direction vector.