Chapter 9, Problem 47RE

To determine

To find : The parametric equations for the line that passes through $P$ and $Q$ .

Explanation of Solution

Given: The points are $P\left(5,2,8\right)$ and $Q\left(4,1,-2\right)$

A line passing through the point $P\left(x_0,y_0,z_0\right)$ and parallel to the vector $v=\langle a,b,c\rangle$ is described by the equations,

$\begin{array}{l}x=x_0+at\\ y=y_0+bt\\ z=z_0+ct\end{array}$

The vector $v$ can be calculated as,

$\begin{array}{c}v=PQ\\ =Q-P\\ =\langle -2,3,1\rangle \end{array}$

Thus, the parametric equations are,

$\begin{array}{c}x=6-2t\\ y=-2+3t\\ z=-3+t\end{array}$

Therefore, the parametric equations are listed above.