2.5 Equations of lines and planes in space  (Page 2/19)

The parametric equations of a line are not unique. Using a different parallel vector or a different point on the line leads to a different, equivalent representation. Each set of parametric equations leads to a related set of symmetric equations, so it follows that a symmetric equation of a line is not unique either.

Sometimes we don’t want the equation of a whole line, just a line segment. In this case, we limit the values of our parameter $t.$ For example, let $P\left(x_0,y_0,z_0\right)$ and $Q\left(x_1,y_1,z_1\right)$ be points on a line, and let $\text{p}=⟨x_0,y_0,z_0⟩$ and $\text{q}=⟨x_1,y_1,z_1⟩$ be the associated position vectors. In addition, let $\text{r}=⟨x,y,z⟩.$ We want to find a vector equation for the line segment between $P$ and $Q.$ Using $P$ as our known point on the line, and $\overrightarrow{PQ}=⟨x_1-x_0,y_1-y_0,z_1-z_0⟩$ as the direction vector equation, [link] gives

Using properties of vectors, then

Thus, the vector equation of the line passing through $P$ and $Q$ is

Remember that we didn’t want the equation of the whole line, just the line segment between $P$ and $Q.$ Notice that when $t=0,$ we have $r=p,$ and when $t=1,$ we have $r=q.$ Therefore, the vector equation of the line segment between $P$ and $Q$ is

Going back to [link] , we can also find parametric equations for this line segment. We have

Then, the parametric equations are

Distance between a point and a line

We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. When building a home, for example, builders must consider “setback” requirements, when structures or fixtures have to be a certain distance from the property line. Air travel offers another example. Airlines are concerned about the distances between populated areas and proposed flight paths.