Exercise 23

Give the parametric form of a line that goes through the points $(1,2,3)$ and $(4,5,6)$.

Do the lines given by $\overrightarrow{r}_1(t)=\langle 0,1,1 \rangle + t\langle 1,2,2 \rangle$ and $\overrightarrow{r}_2(t)=\langle 2,0,3 \rangle + t\langle 1,4,4 \rangle$ intersect? If so, state at what point(s) they intersect.

Give a parametric form of the y-axis.

Give a parametric form of a line that does not intersect the xz-plane.

Give a parametric form for the line that goes through points given by $x=2y=3z$.