Define a Line using Vectors

Let's start with a problem: 

Write the equation of a line containing the points (2,3,4) and (4,5,9).

We can start by writing the "slope", which is just the vector between the points, which is $<2,2,5>$. It's important to note that this line is parallel to the vector $<2,2,5>$, so we're not done yet. We can say the line equals $(2, 3, 4) + t<2,2,5>$. Now, if we combine this into one vector, it's $ <2 + 2t, 3 + 2t, 4 + 5t> $. That's called the vector form. The parametric form is made by splitting the vector up into its separate parts. $$\text{Vector form: }$$ $$r = (2,3,4) + t\cdot <2,2,5>$$ $$\text{Parametric form: }$$ $$x(t) = 2 + 2t $$ $$y(t) = 3 + 2t $$ $$z(t) = 4 + 5t $$

Write the equation of a line parallel to the previous line, but containing the point (17, 2, 3).

We know the "slope" is the same, and it contains the point $(17,2,3)$. That's literally all we need to know. $$\text{Vector form: } (17,2,3) + t<2,2,5> \text{ or } <17 + 2t, 2 + 2t, 3 + 5t> $$ $$\text{Parametric form: } \begin{cases} x(t) = 17 + 2t \\ y(t) = 2 + 2t \\ z(t) = 3 + 5t \end{cases}$$