Answer:
[tex]\vec{r}(t)=\left<-1,\frac{1}{2},4+t\right>[/tex]
In parametric form:
[tex]x=-1, y=\displaystyle\frac{1}{2}, z=4+t[/tex]
Step-by-step explanation:
We first find the mid point:
[tex]\displaystyle\left(\frac{0+(-2)}{2},\frac{-3+4}{2},\frac{7+1}{2}\right)=\left(-1,\frac{1}{2},4\right)[/tex]
Then, a vector parallel to the z-axis is:
[tex]\vec{v}=\left< 0,0,1 \right>[/tex]
Then remember the equation of a line passing through point [tex](x_o,y_o,z_o)[/tex] and parallel to the vector [tex]\vec{v}=\left<a,b,c\right>[/tex] is:
[tex]\vec{r}(t)=\left<x_o,y_o,z_o\right>+t\left<a,b,c\right>[/tex]
So, in this problem we get:
[tex]\vec{r}(t)=\left<-1,\frac{1}{2},4\right>+t\left<0,0,1\right>[/tex]
After combining the two vectors:
[tex]\vec{r}(t)=\left<-1,\frac{1}{2},4+t\right>[/tex]
In parametric form:
[tex]x=-1, y=\displaystyle\frac{1}{2}, z=4+t[/tex]
