1. **State the problem:** Find the equation of the line passing through points $A(4,1,2)$ and $B(6,-3,4)$.\n\n2. **Formula and rules:** The vector equation of a line through 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.\n\n3. **Find the direction vector:** $$\mathbf{d} = \overrightarrow{AB} = B - A = (6-4, -3-1, 4-2) = (2, -4, 2)$$\n\n4. **Write the vector equation:** Using point $A(4,1,2)$ and direction vector $\mathbf{d} = (2,-4,2)$: $$\mathbf{r} = (4,1,2) + t(2,-4,2)$$\n\n5. **Parametric form:** $$x = 4 + 2t$$ $$y = 1 - 4t$$ $$z = 2 + 2t$$\n\nThis is the equation of the line passing through points $A$ and $B$.