# Find a vector equation for the line segment from (2, 1,4) to (4,6,1)

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???

## 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???

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Learn mathKrista KingMarch 7, 2020math, learn online, online course, online math, calc 3, calculus 3, calc iii, calculus iii, vector equation of a line segment, parametric equation of a line segment

### What is a vector equation example?

Example 18.6 The vector equation of a line between two points a and b was found in Chapter 9 to be r = a(l – t)+ b t where t is some parameter and for points between A and B then 0≤t ≤ 1.

### What is the formula for a line segment?

Ans) d =√(x2– x1)2+(y2 – y1)2, or line segment is a distance between two points, so we can also use the distance formula.