Question

Find the vector equation of the line through ​(0​,0​,0​) and ​(5​,1​,2​) where tequals0 corresponds to the first given point and where tequals1 corresponds to the second given point.

Answer 1

The equation of a line passing through the points (0,0,0) and (5,1,2) is:
$\frac{x-0}{5-0} = \frac{y-0}{1-0} = \frac{z-0}{2-0}$;
$\frac{x}{5} = \frac{y}{1} = \frac{z}{2}$.

Take $\frac{y}{1} =t$, then $\frac{x}{5}=t \\ \frac{y}{1}=t \\ \frac{z}{2}=t$ $\rightarrow$ $x=5t \\ y=t \\ z=2t$. 
This is parametrical equation of a straight line, where $y|_{t=0}=0 \\ y|_{t=1}=1$.