Step 1: A Vector Equation Route:
First write the line in vector form, then convert it to Cartesian form by comparing coefficients.
The vector equation of a line through point $\vec a$ and parallel to vector $\vec b$ is $\vec r=\vec a+t\vec b$, where t is a parameter.
Step 2: Write the Vector Equation:
Here $\vec a=5\hat i+2\hat j-4\hat k$ and $\vec b=3\hat i+2\hat j-8\hat k$.
\[ \vec r=(5\hat i+2\hat j-4\hat k)+t(3\hat i+2\hat j-8\hat k) \]
Step 3: Extract Coordinates and Eliminate t:
Writing $\vec r=x\hat i+y\hat j+z\hat k$ gives three scalar equations.
\[ x=5+3t,\quad y=2+2t,\quad z=-4-8t \]
Solve each equation for t.
\[ t=\dfrac{x-5}{3},\quad t=\dfrac{y-2}{2},\quad t=\dfrac{z+4}{-8} \]
Since all three expressions equal the same t, they must equal each other.
Final Answer:
Equating the three expressions for t gives the Cartesian equation of the line.
\[ \boxed{\dfrac{x-5}{3}=\dfrac{y-2}{2}=\dfrac{z+4}{-8}} \]