# How do you find the parametric equations of line through the origin and parallel to the line determined by x = 2t, y = 1 - t, z = 4 + 3t?

Jan 1, 2017

$m a t h b f r = t \left(\begin{matrix}2 \\ - 1 \\ 3\end{matrix}\right)$

#### Explanation:

The line is:

$m a t h b f r = \left(\begin{matrix}2 t \\ 1 - t \\ 4 + 3 t\end{matrix}\right) = \left(\begin{matrix}0 \\ 1 \\ 4\end{matrix}\right) + t \left(\begin{matrix}2 \\ - 1 \\ 3\end{matrix}\right)$

And so has direction vector:

$m a t h b f d = \left(\begin{matrix}2 \\ - 1 \\ 3\end{matrix}\right)$.

The line through the Origin with this direction can be written as:

$m a t h b f r = m a t h b f 0 + t m a t h b f d = t \left(\begin{matrix}2 \\ - 1 \\ 3\end{matrix}\right)$