<< Chapter < Page | Chapter >> Page > |
A line $L$ parallel to vector $\text{v}=\u27e8a,b,c\u27e9$ and passing through point $P\left({x}_{0},{y}_{0},{z}_{0}\right)$ can be described by the following parametric equations:
If the constants $a,b,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c$ are all nonzero, then $L$ can be described by the symmetric equation of the line:
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.
Find parametric and symmetric equations of the line passing through points $\left(1,4,\mathrm{-2}\right)$ and $(\mathrm{-3},5,0).$
First, identify a vector parallel to the line:
Use either of the given points on the line to complete the parametric equations:
Solve each equation for $t$ to create the symmetric equation of the line:
Find parametric and symmetric equations of the line passing through points $\left(1,\mathrm{-3},2\right)$ and $\left(5,\mathrm{-2},8\right).$
Possible set of parametric equations: $x=1+4t,y=\mathrm{-3}+t,z=2+6t;$
related set of symmetric equations: $\frac{x-1}{4}=y+3=\frac{z-2}{6}$
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 $\text{p}=\u27e8{x}_{0},{y}_{0},{z}_{0}\u27e9$ and $\text{q}=\u27e8{x}_{1},{y}_{1},{z}_{1}\u27e9$ be the associated position vectors. In addition, let $\text{r}=\u27e8x,y,z\u27e9.$ 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 $\overrightarrow{PQ}=\u27e8{x}_{1}-{x}_{0},{y}_{1}-{y}_{0},{z}_{1}-{z}_{0}\u27e9$ 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
Find parametric equations of the line segment between the points $P(2,1,4)$ and $Q\left(3,\mathrm{-1},3\right).$
By [link] , we have
Working with each component separately, we get
and
Therefore, the parametric equations for the line segment are
Find parametric equations of the line segment between points $P(\mathrm{-1},3,6)$ and $Q\left(\mathrm{-8},2,4\right).$
$x=\mathrm{-1}-7t,y=3-t,z=6-2t,0\le t\le 1$
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.
Notification Switch
Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?