Answer
Vector form:$\begin{bmatrix}{x \\y \\z}\end{bmatrix} = t\begin{bmatrix}{1 \\-1\\4}\end{bmatrix}$
Parametric form: $x=t, y=-t,z=4t$
Work Step by Step
The vector form of a line is: $x=p+td$
This implies, $\begin{bmatrix}{x \\y \\z}\end{bmatrix} = \begin{bmatrix}{0 \\0 \\0}\end{bmatrix}+t\begin{bmatrix}{1 \\-1\\4}\end{bmatrix}$
or, $\begin{bmatrix}{x \\y \\z}\end{bmatrix} = t\begin{bmatrix}{1 \\-1\\4}\end{bmatrix}$
And
Parametric equations of a line are defined as such equations which correspond to the components of the vector.
Thus, the parametric form of the equation of a line is:
$x=t, y=-t,z=4t$
Hence, the vector form is:$\begin{bmatrix}{x \\y \\z}\end{bmatrix} = t\begin{bmatrix}{1 \\-1\\4}\end{bmatrix}$
Parametric form: $x=t, y=-t,z=4t$
