The vector and parametric equations of a line segment
The relationship between the vector and parametric equations of a line segment
Sometimes we need to find the equation of a line segment when we only have the endpoints of the line segment.
The vector equation of the line segment is given by
???r(t)=(1-t)r_0+tr_1???
where ???0\le{t}\le1??? and ???r_0??? and ???r_1??? are the vector equivalents of the endpoints.
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The parametric equations of the line segment are given by
???x=r(t)_1???
???y=r(t)_2???
???z=r(t)_3???
where ???r(t)_1???, ???r(t)_2??? and ???r(t)_3??? come from the vector function
???r(t)= r(t)_1\bold i+r(t)_2\bold j+r(t)_3\bold k???
???r(t)=\langle{r}(t)_1,{r}(t)_2,{r}(t)_3\rangle???
Given the endpoints of a line segment, we’ll build the vector equation first, then pull the parametric equations from the vector equation
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Finding vector and parametric equations from the endpoints of the line segment
Example
Find the vector and parametric equations of the line segment defined by its endpoints.
???P(1,2,-1)???
???Q(1,0,3)???
To find the vector equation of the line segment, we’ll convert its endpoints to their vector equivalents.
???P(1,2,-1)??? becomes ???r_0=\langle1,2,-1\rangle???
???Q(1,0,3)??? becomes ???r_1=\langle1,0,3\rangle???
Plugging these into the vector formula for the equation of the line segment gives
???r(t)=(1-t)\langle1,2,-1\rangle+t\langle1,0,3\rangle???
???r(t)=\langle1-t,2-2t,-1+t\rangle+\langle{t},0,3t\rangle???
???r(t)=\langle1-t+t,2-2t+0,-1+t+3t\rangle???
???r(t)=\langle1,2-2t,-1+4t\rangle???
To find the vector equation of the line segment, we’ll convert its endpoints to their vector equivalents.
We can also write the vector equation as
???r(t) = 1\bold i +(2-2t)\bold j +(-1+4t)\bold k???
???r(t) = \bold i +(2-2t)\bold j +(-1+4t)\bold k???
Now that we have the vector equation of the line segment, we just take its direction numbers, or the coefficients on ???\bold i???, ???\bold j??? and ???bold k??? to get the parametric equations of the line segment.
???x=1???
???y=2-2t???
???z=-1+4t???
We’ll summarize our findings.
The vector equation is ???r(t)=\langle1,2-2t,-1+4t\rangle???
The parametric equation is given by ???x=1???, ???y=2-2t???, and ???z=-1+4t???
