1. **State the problem:** Convert the Cartesian equation of the line $$\frac{x - 5}{3} = \frac{y + 4}{7} = \frac{z - 6}{2}$$ into its vector form. 2. **Recall the formula:** The vector form of a line passing through a point $\mathbf{r_0} = (x_0, y_0, z_0)$ with direction vector $\mathbf{d} = (a, b, c)$ is: $$\mathbf{r} = \mathbf{r_0} + \lambda \mathbf{d}$$ where $\lambda$ is a scalar parameter. 3. **Identify the point and direction vector:** From the given equation, - Point on the line: $\mathbf{r_0} = (5, -4, 6)$ - Direction ratios: $(3, 7, 2)$, so direction vector $\mathbf{d} = 3\mathbf{i} + 7\mathbf{j} + 2\mathbf{k}$ 4. **Write the vector form:** $$\mathbf{r} = (5, -4, 6) + \lambda (3, 7, 2)$$ or in unit vector notation: $$\mathbf{r} = 5\mathbf{i} - 4\mathbf{j} + 6\mathbf{k} + \lambda (3\mathbf{i} + 7\mathbf{j} + 2\mathbf{k})$$ 5. **Explanation:** This form expresses every point on the line as starting from the fixed point $(5, -4, 6)$ and moving in the direction of the vector $(3, 7, 2)$ scaled by $\lambda$. **Final answer:** $$\boxed{\mathbf{r} = (5, -4, 6) + \lambda (3, 7, 2)}$$