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.

 

[baba89]

I can understand your confusion about vector equations. Let me try to clarify the concept for you.

Firstly, the vector equation r(t) = P1 + t*v represents a line passing through point P1 in the direction of vector v. This equation does not represent all the lines passing through point P1, but only the specific line in the direction of vector v. Think of it as a specific path or trajectory that starts at point P1 and follows the direction of vector v.

Secondly, you are correct in saying that r(t) only passes through point P1 when t = 0. But this does not mean that the equation is not valid. It simply means that at that specific point, the position vector (r(t)) has the same coordinates as the point P1. As t increases, the position vector will move along the direction of vector v, giving us different points on the line.

So, in summary, the vector equation r(t) = P1 + t*v does represent a line passing through point P1 in the direction of vector v. It is important to note that this equation is not the only way to represent a line, but it is a useful tool in understanding the direction and trajectory of a line in vector form.

I hope this helps in your understanding of vector equations. If you have any further questions, please feel free to ask.