Need help understanding vector equations (very basic)

[astonmartin]

If I wanted to find a vector equation for a line that passes through point P1 (x, y) in the direction of vector v <v1, v2>, I would use the equation: r(t) = P1 + t*v

My question is, r(t) is not actually the line that passes through point P1 with "slope" v, is it? Doesn't r(t) [a position vector] represent many different lines that pass through the origin and some point on the line we are trying to find?

In fact, the only time r(t) actually passes through P1 would be when t = 0, right? So we haven't really found an equation for the line P1 (x, y) in the direction of vector v <v1, v2>...have we?

 

[Kurdt]

r(t) is the parametric representation of the line that passes through point P1 in the direction of the vector v. As t varies you add multiples of the vector v to the line. Since the line travels through a defined point with specific direction there is only one of them.