# How do you find a vector parametric equation for the line through the points P= (0,4,-3) and Q=(-5,4,1)?

Jan 15, 2017

$\vec{r} = \left(\begin{matrix}0 \\ 4 \\ - 3\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}- 5 \\ 0 \\ 4\end{matrix}\right)$

#### Explanation:

In vector notation we have;

We have

$\vec{O P} = \left(\begin{matrix}0 \\ 4 \\ - 3\end{matrix}\right)$ and $\vec{O Q} = \left(\begin{matrix}- 5 \\ 4 \\ 1\end{matrix}\right)$

And so the direction of the line is given by:

$\vec{P Q} = \left(\begin{matrix}- 5 \\ 4 \\ 1\end{matrix}\right) - \left(\begin{matrix}0 \\ 4 \\ - 3\end{matrix}\right) = \left(\begin{matrix}- 5 \\ 0 \\ 4\end{matrix}\right)$

And so the vector equation of the line is

$\vec{r} = \left(\begin{matrix}0 \\ 4 \\ - 3\end{matrix}\right) + l a m \mathrm{da} \left(\begin{matrix}- 5 \\ 0 \\ 4\end{matrix}\right)$