vector joining two points in 3d significance

Question

Suppose there are two points in Three dimensional system.Then we can represent the two coordinates by their Position vector and joining them will represent another vector . What would the resulting vector represent....Any coordinate of a point(but isn't every point in 3D represented by position vector) or it serves as the direction for any line line passing through those two points.Can somebody pls explain this to me.

Answer 1

Well if you have two points $p_1$ and $p_2$ whose position vectors are $\mathbf{r}_1$ and $\mathbf{r_2}$ rspectively, then the vector connecting the two of them from $p_2$ to $p_1$ is just the difference $\mathbf{r}_1 - \mathbf{r_2}$. I'm not sure I understand your question, but you have to understand that there is a difference between a point in a 3d space and the position vector from the $origin$ to the point. The vector represents all possible points from the origin to the desired point, so the difference $\mathbf{r}_1 - \mathbf{r_2}$ is really the oriented combination of all the points from $p_2$ to $p_1$, but does not start from the origin.

Answer 2

I like to think about this visually,

Take a note and draw the x-axis and y-axis (after understand 2 dimensions 3 dimensions is the same), then draw 2 vectors from $(0,0)$ to wherever you want, I'll call them $\vec u,\vec v$. Now the vector from the end of $\vec u$ to the end of $\vec v$ is $\vec u-\vec v$. In 3 dimensions it is exactly the same